II 



■n 



HUB 

Mill 

iHI 
HH 

mHHsh 

■Hh 

MMfflni 

■H Pi 

m 



3 

Mr 



HI 

n 

GnwiuinM 



Ji 

■IP 

JfP 
iBI 

Hi 

iW 



—jSmSSmK 

USSSSm 

jIDpS 
iBBl 




y ..•••. ^ 







r .-»-:/+ 



* J 




r <*o 

























v 



1 f 



\f:Ate\\.J 



<T> - © • * 












: ♦*' ♦♦ v 



i0 



V ..*••. *, 



^. '•• 



'v-i 



W\* \;^-/ v^V V^V 



*v 


















r -^^ V^«V- *^<> +JBIK^\ '"+*<$ o\ 



r *<y 



^ - 










V v «»* >*v 




^^,• , A o 





: ^of 



-^ 



<^ ••-•• s> 



!&• *> " " " V 















<j5°v* 



,o. 









^ ^^ffir° \^^" ;% ^K'- \«f ; ' 






"oV 









*°V, •• 






■• v** ••&*&• ^^' •■ 













V a^.^>.% ./..^i/% 



;• * v V •? 
















* v 















«P^a 



^°«o 



.' ** 



F?E*y "v : ^-\/ v v^^'V V 2 ^^ 















**&k \ <- o .-^fe.^ ***&%> /.'afe. - 



r ^o^ 



•• ^ 






f /V *.V sv ^ 




♦♦*% V 



V •!••- <^ 4*^ r « 



•^^ 









ELEMENTS OF GEODESY 



BY 



J. HOWARD GORE, B.S., 

Professor of Mathematics in The Columbian University ; sometime Astronomer 

and Topographer T T S, Geological Survey; Acting Assistant U. S. 

Coast and Geodetic Survey; Associate des P reus sisc hen 

Geodatischen Institutes. 




* ' - 



NEW YORK: 
JOHN WILEY & SONS, 

15 Astor Place. 



1886. 






\J° 






Copyright, 1886, 
By JOHN WILEY & SONS. 



PREFACE. 



THE chief reason for making the following pages public is 
the desire to put into better shape the principles of Geodesy, 
and have accessible in a single book what heretofore has been 
scattered through many. The advanced student and prac- 
tised observer will find nothing new in this work, and may, 
when accident throws it into their hands, lay it aside with feel- 
ings of disappointment. But it is hoped that the beginner 
will be enabled to get a clear insight into the subject, and feel 
grateful that the discoveries and writings of many have been 
so condensed or elaborated as to make the study of Geodesy 
pleasant. The plan pursued in the discussions that follow is 
to take up each division in its logical order, develop each for- 
mula step by step, and leave the results or conclusion in the 
shape that the majority of writers have considered the best. 
In the text only occasional acknowledgments have been in- 
serted, though at the end of each chapter a list of books will 
be found to which reference has been frequently made. These 
lists are by no means complete, so far as the literature of the 
subject is concerned, but contain the titles of those books 
which were found the most helpful while engaged in self- 
instruction. The compilation of a complete Bibliography is 



IV PREFACE. 

now in hand, forming a part of a History of Geodesy, which 
will be finished in the course of a few years. 

It is a pleasure to record the interest of Mr. Henry Gan- 
nett, Chief Geographer of the U. S. Geological Survey, which 
prompted him to read the manuscript and suggest important 
improvements. 

I desire to acknowledge my obligations to my associate, Pro- 
fessor H. L. Hodgkins, A.M., for the interest he has shown in 
the work, and for his careful revision of the proof-sheets as 
they came from the press. 

I also wish to express my indebtedness to my friend Miss 
Lizzie P. Brown for her suggestions, and for the elimination of 
errors that otherwise would have seriously blemished the work. 
It is hoped that errors do not remain in sufficient number or 
of such size as to impair the clearness or accuracy of the dis- 
cussions that follow. 

When page 102 was written, it was thought that a satisfac- 
tory formula could be procured for the computation referred 
to, but the increasing doubts regarding the coefficient of re- 
fraction have induced me to omit further consideration of the 
subject. 

Washington, July, 1886. 



GEODETIC OPERATIONS, 



CHAPTER I. 

AN HISTORIC SKETCH OF GEODETIC OPERATIONS. 

One of the first problems that suggested itself for solution 
in the intellectual infancy of mankind was: "What is the 
earth, its size and shape?" The possibility of examining the 
constituency of the superficial strata answered with sufficient 
exactness, for the time being, the first part of the question. 
The natural conclusion deducible from daily experience and 
observation is : were the earth deprived of the irregularities 
produced by the valleys and mountains, its surface would be 
a plane. The exact date of the abandonment of this theory 
is unknown. Froriep refers to a Sanskrit manuscript contain- 
ing the following sentence: "According to the Chaldeans, 
4000 steps of a camel make a mile, 66-f miles a degree, from 
which the circumference of the earth is 24,000 miles." Of the 
authenticated announcements of hypotheses, Pythagoras was 
the first to declare that the earth is spherical. This honor 
is sometimes assigned to Thales and Anaximander. Archi- 
medes gave as an approximate value for the circumference 
300,000 stadia. To Eratosthenes (B.C. 276) belongs the credit 
of making the initial step towards a determination of the cir- 
cumference. He observed that at Syene, in Southern Egypt, 
an object on the day of summer solstice cast no shadow, while 
1 



2 GEODETIC OPERATIONS. 

at Alexandria the sun made an angle with the vertical equal to 
one fiftieth of a circumference. Considering that Alexandria 
was north of Syene, he reasoned that the entire circumference 
of the earth was 50 times the distance between those places, or 
250,000 stadia; this he afterwards increased to 252,000 stadia. 
The neglect of the sun's diameter in the determination of dec- 
lination, and the false supposition that Alexandria and Syene 
were on the same meridian, introduced considerable inaccura- 
cies in his results, the exact amount of which, however, we can- 
not estimate owing to our ignorance as to the length of the 
stadium. 

About two hundred years later Posidonius determined the 
amplitude of the arc between Rhodes and Alexandria from 
observations on the star Canopus at both places. At Rhodes 
he saw this star, when on the meridian, just visible above the 
horizon, and at Alexandria its altitude at the same time was 
-fe of a great circle. From this he concluded that the circum- 
ference was 48 times the distance these places were apart, or 
48 X 5000 stadia == 240,000 stadia. If we know the latitude 
of two points on the same meridian, the difference will be the 
amplitude of the arc passing through them, and the circumfer- 
ence will bear the same ratio to the length of the arc that its 
amplitude bears to four right angles. 

Letronne has shown that the amplitude of the arc Posi- 
donius used is only 5 = -^ of a great circle, and Strabo gives 
4000 stadia as the length of the arc, making the circumference 
288,000 stadia. 

Ptolemy in the second century gave 180,000 stadia for the 
circumference, but does not state his authority. Posch infers 
that it was taken from the Chaldean value, since Ptolemy gives 
a Chaldean mile equal to 7-J stadia, and j\ times 24,000 = 
180,000. In 827 an Arabian caliph imposed upon his astrono- 
mers the task of measuring an. arc, and of deducing from it 
the length of the circumference of the earth. 



HISTORIC SKETCH. 3 

Abulfeda in 1322 gave the following description of the 
method employed by them : There were two parties ; one start- 
ing from a fixed point measured a line due north with a rod, 
the other party going due south ; both continuing until the ob- 
served latitudes were found to differ by one degree from that 
of the starting-point. The first party found 56 miles and the 
second 56! miles for a degree. The latter result was accepted, 
its equivalent being approximately 71 English miles. 

This was a great improvement upon the methods of the 
Grecians, who estimated their distances by days' marches of so 
many stadia a day. 

Fernel in 1525 made a measurement for the determination 
of the length of a degree by counting the number of revolu- 
tions made by a wheel of known circumference in going from 
Paris to Amiens. He applied a correction to reduce the broken 
line to a straight one, and the latitude observations were made 
with a 5-foot sector, giving for a degree 365,088 English feet. 
A few years later Father Riccioli made an arc-determination 
in Italy, but it was too short to be of any importance. The 
first attempt to determine the size of the earth by means of 
triangulation was by Willebrord Snellius in 161 5. He measured 
a base-line with a chain between Leyden and Soeterwood, and 
connected it by means of triangles, 33 in number, so as to com- 
pute the distance from Alcmaar to Bergen-op-Zoom. This 
distance he reduced to its equivalent along a meridian, giving 
an arc of i° \\' 05" amplitude, from which he found 55,074 
toises for a degree (a toise being equal to 6.3946 English feet). 
Kastner has shown that the neglect of spherical excess in the 
reduction of these triangles causes an error of nearly a toise. 
In 1722 the measurements were repeated, using for the angle- 
determinations a sector of 5 feet radius ; this second reduction 
gave 57,033 toises for a degree. One can scarcely conceive of 
the amount of labor such an undertaking necessitated at a 
time when there were no logarithmic tables to lighten the work. 



4 GEODETIC OPERATIONS. 

Norwood in 1635 measured with a chain the distance from 
London to York, obtaining for a degree 57,424 toises. 

In the measurement of angles Snellius had sights attached 
to his sector, making a close reading impracticable. 

While the telescope was made use of as early as 1608, no 
one had thought of putting it on an angle-reading instrument 
until Picard, in 1669, placed in the focus of a telescope spider- 
lines to mark the optical axis, which, according to some au- 
thorities, had already been done by Gascoigne in 1640. He 
measured a base-line nearly 7 miles long, and with a sector of 
10 feet radius, to which was attached a telescope, the angles 
were carefully read, until Malvoisine and Amiens were con- 
nected by a chain of triangles. This gave an arc of 1° 22' 58", 
from which he computed 57,060 toises as the length of a 
degree. At this time the effect of aberration and nutation 
were unknown, which, if allowed for, would have shortened 
his arc by 3". However, when his unit of linear measure was 
more accurately compared with the standard it was found to 
be too short, so that when Lacaille revised the work he ob- 
tained the identical result that Picard had previously an- 
nounced. 

The uncertainty of ascertaining the circumference of the 
earth from so short an arc was so keenly felt at this time that 
the extension of this arc both northward and southward was 
undertaken by the Cassini, father and son, Lahire, and Maraldi, 
carrying it from Paris to Dunkirk, and from Paris to Perpignan, 
the entire arc being about 8° 31'. 

The published results of Picard's work were rendered famous 
by endorsing Newton's hypothesis of universal gravitation. 
Newton had attempted to prove this theory by comparing the 
force of gravity on a body at the moon's distance with the 
power required to keep her in her orbit. He used in his com- 
putations the diameter of the earth as somewhat less than 
7000 miles. The result failed to show the analogy he had cbn- 



HISTORIC SKETCH, 5 

ceived ; so he laid aside his theory, so brilliant in conception, so 
lacking in verification. But twenty years later, when Picard's 
length of a degree was made known, increasing the diameter of 
the earth by about a thousand miles, Newton was able to show 
that the deflection of the orbit of the moon from a straight 
line was equivalent to a fall of 16 feet in one minute, the same 
distance through which a body falls in one second at the sur- 
face of the earth. The distance fallen being as the square of 
the time, it followed that the force of gravity at the surface of 
the earth is 3600 times as great as the force which holds the 
moon in her orbit. This number is the square of 60, which 
therefore expresses the number of times the moon is more dis- 
tant from the centre of the earth than we are. If with the 
rude means employed by Picard his errors had not eliminated 
one another, or if their extent had been discovered without 
knowing their compensating character, the undemonstrated 
law of gravitation would have remained as an hypothesis, ce- 
lestial mechanics would have been without the mainspring of 
its existence, and we would now be groping in the darkness of 
an antecedent century. 

Newton also maintained that, owing to the greater centrifugal 
force of the particles at the equator, a meridian section of the 
earth would be an oblate ellipse ; that is, the equatorial axis 
would exceed the polar. If such were the case, the radius of 
curvature would increase in going from the equator towards 
the pole; and as the latitude is the angle formed by the nor- 
mal with the polar axis, if the normal increases, the arc of a 
constant angle must become larger, therefore the oblate hy- 
pothesis requires for verification that the degrees increase in 
going from the equator towards either pole. Consequently the 
results of Cassini's long arc determination were awaited with 
impatience, until 1718, when the announcement was made 
that the northern degree was shorter than the southern ; 
this pleased the French, as it gave them an opportunity to 



6 GEODETIC OPERATIONS. 

again say that the country across the Channel was a " Naza- 
reth from which no good thing could come." A degree of 
the northern arc gave 56,960 toises, and of the southern 
57,098 toises, from which it appeared that the earth was pro- 
late. 

Huygens in 1691 published his theory regarding centrifu- 
gal motion, describing experiments that proved that a rotat- 
ing mass like the earth would have its greater axis perpendicu- 
lar to the axis of rotation. Hence the terrestrial degrees 
increase northward. It was a part of Newton's theory that as 
the polar diameter is less than the equatorial, the force of 
gravity must increase in going towards the pole, and therefore 
a clock regulated by a pendulum would lose time when carried 
towards the equator. When Richer returned in 1672 from the 
Island of Cayenne, where he had been sent to make astronomic 
observations, he found that his clock while at the island lost 
two minutes a day when compared with its rate at Paris, and, 
furthermore, the length of his pendulum beating seconds was 
\\ lines shorter than the Paris seconds pendulum, showing that 
Cayenne was farther than Paris from the centre of the earth. 
A portion of this difference in the lengths of the pendulums 
was supposed to be due to increased counteracting effect of 
centrifugal force nearer the equator, but Newton showed that 
the discrepancy was too great for a spherical globe. Varin 
and Des Hays had a similar experience with pendulums taken 
to points almost under the equator. 

Under the excitement occasioned by this sharp controversy, as 
well as from a desire to know the truth, the French Academy 
decided to submit the problem to a most crucial test by meas- 
uring one arc crossing the equator, and another within the 
polar circle. Knowing the fierce criticism that would be 
brought to bear upon every feature of the work, the partici- 
pants determined to use the most refined instruments and 
most approved methods. In May, 1735, an expedition consist- 



HISTORIC SKETCH. 7 

ing of Godin, Bouguer, De la Condamine, and Ulloa set out for 
Peru. The base was selected near Quito at an elevation of 
nearly 8000 feet above sea-level. Its length was 7.6 miles as 
deduced from a duplicate measurement, made by two parties 
working in opposite directions. The measuring-rods were of 
wood, twenty feet in length, terminated at either end in copper 
tips to prevent wearing by attrition. 

They were laid approximately horizontal, the deviation 
therefrom, being estimated by a plummet swinging over a 
graduated arc. A comparison with a field standard was made 
each day, this standard being laid off from the toise taken from 
Paris, which afterwards became the legal unit in France, and is 
known as the Toise of Peru. The angles of the 33 triangles 
were measured on quadrants of 2 and 3 feet radius ; these 
were so defective, however, that great care was necessary in de- 
termining the instrumental errors and applying them to each 
angle-determination. Twenty observations w r ere made at dif- 
ferent stations for ascertaining the azimuths. 

The amplitude of the arc was found from simultaneous lati- 
tude-observations made at the terminal stations on the same 
star. Realizing that great uncertainties would arise from a 
faulty determination of the amplitude, the latitude-observa- 
tions were made with sectors 12 and 8 feet radius, on the sup- 
position that the larger the sector the more accurate would be 
the results. But the instability of the supports allowed such 
great flexure that they w T ere almost wholly reconstructed on 
the field. 

A southern base was measured as a check near Cotopaxi 
at an elevation of nearly 10,000 feet above sea-level. Its 
length, 6.4 miles, as measured, differed from the value com- 
puted from the northern base by only one toise, and the entire 
arc was but ten toises longer according to Condamine than 
found by Bouguer. The amplitude as deduced by Bouguer 
was 3 f 1", giving for the length of a degree reduced to sea- 



8 GEODETIC OPERATIONS. 

level 56,753 toises — the mean of the two computations just 
quoted. The field-work occupied two years, but the results 
were not published until the beginning of 1746. 

Von Zach revised the calculations, finding the arc to be 71 
toises shorter ; and Delambre recomputed the latitudes, from 
which he found the amplitude increased by a little more than 
2 seconds. According to the former, a degree would have at 
that latitude a length of 56,731 toises, while the latter would 
give 56,737 toises, a value indorsed by Arago. 

The polar party, consisting of Maupertuis, Clairault, Camus, 
Le Monnier, Outhier, and Celsius, Professor of Astronomy at 
Upsal, reached its destinaton May 21, 1736. The river Tornea, 
flowing south, with mountains of greater or less elevation on 
each side, afforded in its valley a suitable location for the base, 
and the mountains, points for the triangle stations. The 
signals were built of trees stripped of their bark, in the shape 
of a hollow cone. The angles were measured with a quadrant 
of 2 feet radius provided with a micrometer, each angle being 
read by more than one person, the average of the means of the 
individual results being taken. Great care was exercised in cen- 
tring the instrument and in checking the readings by observ- 
ing additional angles whose sums or differences would give the 
angles wanted. 

Latitude-observations were made by determining the differ- 
ence of zenith distances of two stars with a sector consisting of 
a telescope 9 feet long, which formed the radius of an arc 5 3c/. 
This arc was divided into spaces of y' 30", which were subdi- 
vided by a micrometer. From the observations corrected for 
aberration, nutation, and precession, the amplitude was found 
to be 57' 26". 93 according to Outhier, 57' 28^.75 according to 
Maupertuis, and 57' 28".5 as given by Celsius. The base was 
measured during the winter over the frozen snow and ice on 
the river Tornea, the terminal points only being on land. The 
measuring-bars were of wood, each 30 feet long, as determined 



HISTORIC SKETCH. 9 

by comparison with an iron toise carried from Paris. Daily 
comparisons were made by placing the rods between two iron 
nails, previously driven at a distance apart just equal to the 
length of one of the rods on the first day. It was found that 
they had not changed in length during the work. 

There were two parties, each having four rods, which they 
placed end to end on the snow. In this manner the entire 
base was measured twice, both parties laying the same number 
of bars each day giving a daily check. The total difference in 
the two results was only 4 inches in a distance of 8.9 miles, a 
degree of accuracy that is quite remarkable when it is consid- 
ered that the average temperature was 6 degrees F. below 
zero. From this arc a degree cut by the polar circle was ascer- 
tained to be 57,437 toises. While many precautions were taken, 
the disagreement in the astronomic reductions, and some in- 
strumental errors that were afterwards discovered, caused some 
doubt as to the reliability of the work. If correct, a degree at 
this point would be 377 toises longer than a degree at Paris, a 
difference greater than the theorists had calculated, and more 
confirmatory of the oblate hypothesis than was wanted. 

Cassini, De Thuri, and Lacaille revised the French arc pre- 
viously measured by J. and D. Cassini, and, comparing the 
northern with the southern portion of the arc, they declared 
that the earth was oblate ; this was announced in 1744. In 
1743, Clairaut, reasoning that the earth, instead of being of uni- 
form density, each particle being pressed down by all that is 
above, those near the centre must be denser than those nearer 
the surface. Starting with the hypothesis that the density is 
a function of the distance from the surface, he declared that 
the earth was oblate, but not to the extent that Newton had 
supposed. 

Let us, in review, contemplate the condition of this problem 
at this period : Newton, in 1687, from a theoretic analysis, said 
the earth was oblate; this explained the behavior of Richer's 



IO GEODETIC OPERATIONS. 

clock in 1672. Huygens, in 1691, revolved a hollow metallic globe, 
and saw it protrude at the centre ; hence, from analogy, he ac- 
cepted the oblate hypothesis. Cassini's arc of 1718 declared 
the theorists wrong. The Lapland labors of Maupertuis, nine- 
teen years later, negatived Cassini's conclusion. Clairaut, in 
1743, endorsed Maupertuis, but failed to show so great an ob- 
lateness. In 1744, Lacaille, repeating the work of Cassini, 
changed the results until they conformed to theory; and hardly 
a year later came the fruit of the ten years' labor in Peru to as- 
sert that Newton, Huygens, and Clairaut were all right, in dif- 
ferent degrees. 

Lacaille, in 1750, went to the Cape of Good Hope to deter- 
mine the moon's parallax, and while there he measured an arc 
of 1 J degrees in south latitude 33 18^', from which he deduced 
57,037 toises as the length of a degree. The short time de- 
voted to this work, and the inferior quality of his instruments, 
caused this determination to be lightly regarded. The next 
triangulation was executed by Boscovich in 1751-53, in latitude 
43 N., where an arc of 2° gave 56,973 toises as the length of 
a degree. In 1768 Beccaria found 57,024 toises for a degree 
in latitude 44 44' N. Zach revised this work and found a dif- 
ference of 15 toises in the length of the arc, and numerous 
errors in the angle-measurements. Also the proximity of the 
northern terminus of the arc to the mountains suggests that 
the unnoticed deflection of the plumb-line gave to the arc a 
wrong amplitude. 

In connection with Liesganig, the indefatigable Boscovich 
measured an arc of 3 , giving for the northern portion in lati- 
tude 48 43', 57,086 toises for a degree, and for the southern 
part they found a degree to be 56,88 1 toises — a difference too 
great to give to the work much confidence. 

The surveyors Mason and Dixon (1764-68), in locating the 
boundary-line between the properties of the Penn family and 
Lord Baltimore, a portion of which afterwards became the 



HISTORIC SKETCH. II 

boundary-line between Pennsylvania and Maryland, saw that 
that part of the line separating Maryland from Delaware was 
located on low and level land, almost coinciding with a merid- 
ian. For this reason they concluded that it would be suitable 
for measuring the length of a degree. The Royal Society of 
London voted them money for the work. The whole distance 
was measured with wooden rods 20 feet in length ; contact was 
carefully made with rods level, and thermometric readings 
made to correct for expansion. Latitude was ascertained 
from equal zenith-distance observations, and azimuth meas- 
ured from a meridian mark determined from astronomic obser- 
vations. 

The amplitude of the arc was i°28 / 45 // , an d the length as 
measured gave for a degree 56,888 toises. 

In 1783 the proposition was made on the part of the French 
geodesists to unite Paris and Greenwich by triangulation. 
General Roy was placed in charge of the operations on the 
English side of the Channel, and Count Cassini, Mechain, and 
Legendre attended to that part of the work that fell within 
France. In this work every precaution was taken to secure 
good results, and all refinements at that time devised were 
utilized. For the first time Ramsden's theodolite with a circle 
of 3 feet in diameter was employed in measuring the angles. 

This circle was divided into 15-minute spaces, and was read 
at three points by micrometers rigidly connected with one 
another. The telescope had a focal length of three feet, and of 
sufficient power to render visible a church-tower at a distance 
of forty-eight miles across water. The history of this the- 
odolite would form a large part of the history of the English 
triangulation. Sir Henry James, in speaking of it in 1863, 
said : " When it is considered that this instrument has been 
in use for the last seventy-five years, and that it has been 
placed upon many of our very highest mountains, on our most 
distant islands, and on the pinnacles of our loftiest churches, the 



12 GEODETIC OPERATIONS. 

perfection with which this instrument was made, and the care 
with which it has been preserved, is truly remarkable." Also 
Colonel Clarke, in 1880, remarks that it is as good as when it 
left the workshop. 

The triangulation in England rested upon the Hounslow 
Heath base. The first measurement of this base was made in 
June, 1784, with a steel chain of 100 feet in length, giving for 
the length of the line, corrected for temperature, 27,408.22 feet. 
A second determination was made using wooden rods, termi- 
nating in bell-metal tips, the entire length being 20 feet 3 inches. 
In the course of the work it was noticed that the rods were 
affected by moisture so as to render the results, 27,406.26 feet, 
unreliable. At the suggestion of Colonel Calderwood, it was 
decided to measure the line with glass tubes. These were 20 
feet long, supported in wooden cases 8 inches deep, and con- 
tact was made as in the slide-contact forms. In the reduction 
of the length of the base a carefully determined coefficient of 
expansion, .0000043, was employed, giving for the length of the 
base 27,404.0137 feet. 

Another measurement made with a steel chain, using five 
thermometers for temperature-indications, gave a result differ- 
ing from the last by only 2 inches. This length was the equiva- 
lent reduced to sea-level — a correction being applied for the 
first time in the history of geodesy. 

In the French work nothing new was introduced except the 
repeating-circle. This was constructed on a principle pointed 
out by Tobias Mayer, Professor in the University of Gottin- 
gen, which was thought to eliminate errors of graduation that 
had at that time become a source of fear, owing to the imper- 
fect means for graduating. By the method of repetition it 
was supposed that if a number of pointings be made with 
equal care, and the final reading be divided by the number of 
pointings, the error of graduation as affecting the angle so re- 
peated would be likewise divided, and hence be too small to 



HISTORIC SKETCH. 1 3 

be appreciable. If all the parts of the instrument were rigid, 
and if the circle or telescope could be clamped in place without 
the one in its motion moving the other, the theory might be 
endorsed in practice. However, these conditions have never 
been definitely secured, nor is it likely that a clamp can be de- 
vised that will not give in its working a travelling motion. 
These obstacles did not present themselves with sufficient 
force to cause the French to abandon this form of angle-read- 
ing instruments until it had mutilated their labors covering 
a half-century. 

Barrow, in 1790, measured an arc of i° 8' in East Indies, ob- 
taining for a degree in latitude 23 18", 56,725 toises. 

The year 1791 carries with it the honor of having witnessed 
the inception of the most majestic scheme ever devised for ob- 
taining and fixing a standard unit of measure. Laplace and 
Lagrange, with the support of the principal mathematicians of 
that period in France, proposed to the Assembly of France 
that the standard linear unit should be a ten-millionth part of 
the earth's quadrant, to be called a metre ; the length of this 
quadrant to be determined by the measurement of an arc of 
9 40' 24", of which nearly two thirds was north of the 45th 
parallel,— the northern terminus being Dunkirk, and the south- 
ern, Barcelona. Delambre was in charge of the work from 
Dunkirk to Rodez, and Mechain completed that portion ex- 
tending from Rodez to Barcelona. 

Two base-lines were measured, one at Melun, near Paris, and 
the other at Perpignan, each about seven and a quarter miles 
long. The measuring-bars were four in number, each com- 
posed of two strips of metal two toises in length, half an inch 
in width, and a twelfth of an inch in thickness. The two 
metal strips were supported on a stout beam of wood, the 
whole resting on iron tripods provided with levelling-screws. 

One of the strips was made of platinum ; the other, resting on 
this, was copper, sh-orter than the platinum by about 6 inches. 



14 GEODETIC OPERATIONS. 

At one end they were firmly fastened together, but free to 
move throughout the remainder of their lengths ; so that by 
means of a graduated scale on the free end of the copper and 
a vernier on the corresponding end of the platinum, the vary- 
ing lengths owing to the different expansions of the two 
metals could be determined, and hence the temperature known. 
This was the invention of Borda, and is now known as the 
Borda scale, or metallic thermometer. The bars were compared 
indirectly with the toise of Peru by their maker, and No. I of 
this set afterwards became a standard of reference. The angles 
were measured with repeating-theodolites, and azimuth was 
determined at five principal stations by measuring the angle 
between another station and the sun, mornings and even- 
ings. Latitudes were computed from zenith-distance observa- 
tions at the termini and at three intermediate points. A com- 
mission was appointed to review all the calculations : they 
combined this arc with the Peruvian, deducing the length of a 
quadrant whose legalized fractional part is the present metre. 

Nouet, while astronomer to the French expedition to Africa 
in 1798, measured a short arc, from which he found a degree to 
be 56,880 toises. The disagreement between the computed 
and observed azimuths obtained by Maupertuis — amounting to 
34" in the terminal line — caused considerable suspicion to attach 
to the entire work. The Stockholm Academy of Sciences de- 
cided to have the stations reoccupied, and consequently, in 
1 801, sent Svanberg, Palander, and two others to Lapland for 
that purpose. They did not recover all of the previously oc- 
cupied stations, nor did they use the same terminal points, but 
deduced as an independent value for a degree 57,196 toises. 
' Major Lambton measured an arc of i°33 / 56 // in India in a 
mean latitude of 12° N. in 1802. After his death, in 1805, it 
was continued by Colonel Everest with such vigor that by 1825 
an arc of 16 was completed. 

The French gave the English an impetus to push forward 



HISTORIC SKETCH. 1 5 

geodetic work by their co-operation in the connection already 
referred to, so that while in England a trigonometric survey was 
being prosecuted, the requisite care was bestowed upon it to 
make it of value in degree-determinations. From 1783 to 1800 
this survey was under the direction of General Roy. Mudge 
continued the triangulation for two years, completing an arc of 
2° 50', from which he found for the length of a degree in lati- 
tude 53 , 57,017 toises, and in 51 , 57,108 toises ; therefore the 
degrees shorten towards the pole. 

Mechain wished to carry his arc south of Barcelona to the 
Balearic Isles, but was prevented by his unfortunate death. 
However, the energetic mathematicians who made that period 
of the French history so brilliant would not allow such a fea- 
sible project to remain incomplete. So Biot and Arago spent 
two years, beginning in 1806, in extending the triangulation 
from Mt. Mongo, on the coast of Valencia, to Formentera, giv- 
ing a complete arc of 12° 22' l3' / -44. 

The latitude of Formentera was determined from nearly 
4000 observations on a and fi Ursae Minoris, but owing to the 
fact that they were all made on stars on one side of the zenith, 
erroneous star-places would introduce serious errors in the re- 
sulting latitude, as demonstrated by Biot in 1825, when he ob- 
tained for that station a latitude differing by 9" from the first. 
The length of a degree as published in 1821 was 57,027 toises 
in latitude 45 N. Bessel, using the corrected latitude of For- 
mentera, found 56,964 toises ; and in 1 841 Puissant discovered 
another error which changed the degree's length to 57,032 
toises. In the reduction of this work the principal of least 
squares was used for the first time in adjusting the triangulation 
in conformity with the geometric conditions, as will be explained 
in a future chapter. 

The errors already referred to in the reduction of this work 
show the fallacy of accepting any determination of the earth's 
quadrant as an unvarying quantity from which a standard, if 



1 6 GEODETIC OPERATIONS. 

lost or destroyed, could be definitely restored with a length 
identical with the previous one. Even if the earth be perfectly 
fixed and stable in its size and shape, of which there is great 
doubt, and the ten-millionth part of a quadrant always the 
same, the uncertainties in obtaining the same value for this 
quadrant twice in succession outweigh the utility of the plan 
and the majesty of its conception. This is not intended as an 
argument against the decimal feature, or the readiness with 
which units of weight can be obtained from those of volume. 
In this respect the metric system is superior to all others now 
in use, and these advantages alone warrant its universal adop- 
tion, while the fixity of the standards preserved by the Inter- 
national Bureau of Weights and Measures is sufficiently certain 
to dispel all doubts as to the change of length of the metre, 
without feeling the necessity of frequently comparing it with 
a physical law or mass supposed to be immutable. 

Prussia began geodetic work in 1802 with the measurement 
of a base-line near Seeburg by von Zach. This line was care- 
fully measured and the end-points fixedly marked by inclosing 
in masonry iron cannons with the mouth upwards. In the 
mouth a brass cylinder was fastened by having lead run around 
it ; the cross-lines on the upper surface of the cylinders denoted 
the end of the line. The triangulation began in 1805, but was 
stopped by the war with France in 1806, although Gotha, the 
province in which the work was being prosecuted, remained 
neutral. After the battle of Jena the people of Gotha, fear- 
ing that the French would not regard their neutrality lasting, 
especially if they should be suspected of harboring concealed 
weapons, caused these cannons to be dug out and carefully 
hid, thus sacrificing some accurate work to allay a foolish 
fear. 

Under Napoleon I. the importance of faithful maps for war 
purposes at least was keenly felt, and to secure men trained 
for the preparation of such maps the Ingenieur Corps was or- 



HISTORIC SKETCH. 1 7 

ganized, also the Ecole Polytechnique and the Ecole Speciale 
de G£odesie. The basis of an accurate cartographic survey' 
must be a triangulation, and degree-measurements had such a 
strong hold upon the mathematicians that the advisability of 
giving to the triangulation the requisite accuracy to make it 
useful for such determinations was never questioned. 

Switzerland and Italy were to join their work to that of 
France, to give an arc of parallel from the Atlantic Ocean to 
the Adriatic Sea. This was begun in 1811, and continued by 
one or more of the countries until its completion in 1832, giv- 
ing an arc of 12° 59/ 4". Owing to serious discrepancies be- 
tween the observed and computed values, this work received 
but little credit. In one instance the difference in azimuth 
was 49". 5 5, and in longitude the difference between the geo- 
detic and astronomic was 3J_j29- 

The French expedition to Lapland for the purpose of an arc- 
measurement incited the first astronomer of the St. Petersburg 
Academy, De ITsle, to make a similar determination in Russia. 
In 1737 he measured a base-line on the ice between Kronstadt 
and Peterhof, and occupied several stations during that and 
the two following years. However, it came to an end very 
abruptly without leaving any definite results by which to re- 
member it. 

The first geodetic work in Russia that deserves the name 
was begun in 1817 under the patronage of Alexander I., with 
Colonel Tenner and Director Struve at the head. Tenner 
began in the province of Wilma and continued until 1827, by 
which time he had completed an arc of 4^°, using a base 
measured with an apparatus of his own devising, consisting of 
two parallel bars of iron firmly fastened together. The angles 
were read on a 16-inch repeating theodolite. Struve did not 
receive his instruments until 182 1, but in the ten years follow- 
ing he finished an arc of 3-J . 

There was now a gap of about 5-J between the Russian and 
2 



1 8 GEODETIC OPERATIONS. 

the Lapland arcs which it was desired to close up. In this 
work Struve was assisted by Argelander. They measured a 
check-base with Struve's apparatus, completing the entire task 
in 1844. In the mean time Tenner had added 3 25' to his arc. 
Just here it might be of interest to remark that Bessel had 
communicated to Tenner his discussion regarding the figure 
and size of the earth. This was appended to Tenner's manu- 
script record and placed in the care of the St. Petersburg 
Academy in 1834, three years before it was published by Bes- 
sel in the Astronomische Nachrichten, No. 333. 

Permission was obtained from the Swedish authorities to 
continue this arc across Norway and Sweden. This also was 
placed under the direction of Struve, with the assistance of 
Selander and Hansteen. The former finished his share of the 
triangulation with a measured base in 1850. Hansteen com- 
pleted the Norwegian portion, checking on a base of 1 155 
toises. The Russian parties, together with their co-laborers, 
by 1855 had completed a meridional arc of 2 5 20' 9". 29, ex- 
tending from the Danube to the North Sea. Of this there 
were two great divisions — the Russian, with 8 bases and 224 
principal triangles and 9 latitude-determinations ; and the 
Scandinavian, with 2 bases, 33 principal triangles, and 4 astro- 
nomic stations. Prior to 1821 the principle of repetition was 
exclusively used on horizontal circles in its original form. 
Struve then decided that the periodic errors noticed when the 
simple method of repetition was employed could be partially 
eliminated by reversing the direction of rotation ; but he soon 
abandoned this, and in 1822 began to measure angles a number 
of times on different parts of the circle. 

The test of the accuracy of this work is in the difference in 
the lengths of junction-lines as computed from different bases. 
From an examination of ten of these differences, I have found 
that the average is 0.1718 toise, with 0.0179 as the minimum 
and 0.4764 for a maximum. The values found for a degree 



HISTORIC SKETCH. 1 9 

were: 57,092 toises in latitude 53*20', 57,u6 in 55 34', 57,121 
in 56 32, 56,956 in 57° 28', and 57,125 in 59 14'. The utility 
of this arc for degree-measurements is not proportionate to its 
immensity, because of the fewness of the astronomic deter- 
minations — only one in every two degrees of amplitude. 

General von Muffling in 1818 connected the Observatory of 
Seeburg with Dunkirk, and determined the amplitude of the 
arc by measuring the difference in time between the stations 
two by two. This was done by recording in local time the ex- 
act instant at which a powder-flash set off at one station at a 
known local time was seen at the other. The amplitude of 
this arc, embracing 8 determinations of this kind in its chain, 
was8°2i' 18". 

Between 1818 and 1823 Colonel Bonne connected Brest with 
Strasburg, with a base near Plouescat. It is interesting to 
note that in this work angles were measured at night, using 
as a signal a light placed in the focus of a parabolic reflec- 
tor. Differences of longitude were determined by powder- 
flashes. 

Gauss began the trigonometric survey of Hanover in 1820, 
measuring an arc of 2° 57', from which he found for a degree 
57,126 toises in the same latitude in which Mudge in England 
obtained for a degree 57,016 toises, and Musschenbroeck, in 
Holland, 57,033 toises. It was while engaged upon this work 
that Gauss first used the heliotrope that has since borne his 
name. 

Schumacher at the same time commenced the Danish trian- 
gulation with the advice and assistance of Struve. His arc 
of i° 31' 53" gave for a degree 57,092 toises in latitude 54 
8' 13". 

In 1821 Schwerd concluded from his measurement of the 
Speyer base that a short line most carefully measured would 
give as good results as a longer one on which the same time 
and labor would be expended. From his base of 859.44 M. he 



20 GEODETIC OPERATIONS. 

computed the length of Lammle's base of 19,795.289 M., giving 
a difference of only 0.0697 M. 

Colonel Everest was appointed to succeed Colonel Lambton 
in the direction of the great trigonometric survey of India in 
1823. During the following seven years he measured three 
bases with the Colby apparatus as checks to the triangulation 
which he extended from 18 3' to 24 f. To Colonel Everest 
is due the credit of introducing greater care in all the linear 
and angular determinations. In the latter he employed 
the method of directions in greater number than did his pred- 
ecessors. 

In 1 83 1 Bessel and Baeyer undertook a scheme of triangula- 
tion that was to unite the chains of France, Hanover, Den- 
mark, Prussia, and Bavaria with that of Russia, and at the 
same time serve for degree-measurements. It was oblique, so 
that, by determining the direction and amplitude, degrees of 
longitude as well as latitude could be found. The base-line 
near Fuchsburg was measured with a slightly modified form 
of the Borda apparatus now known as Bessel's apparatus, of 
which there is now an exact copy in use in the Landes Trian- 
gulation of Prussia. The length of this base was 934.993 
toises when reduced to sea-level. The ends were marked by a 
pier of masonry inclosing a granite block, in whose top was set 
a brass cylinder carrying cross-lines indicating the end of the 
base. Just above this was built a hollow brick column high 
enough for the theodolite support, with a larger square stone 
for a cap-stone. In the centre of this there was a cylinder 
coaxial with the one below, so that the instrument could be 
placed immediately over the termini of the base. The theodo- 
lites had 12- and 15-inch circles, read by verniers, and the angles 
were read by fixing the zeros coincident, and then turning to 
each signal in succession with verniers read and recorded for 
each. After completing the series, the signals were observed 
in inverse order, the means of the two readings giving a set of 



HISTORIC SKETCH. 21 

directions. The zero would then be shifted to another position, 
and all the signals sighted both in direct and inverted order, 
until a desired number of sets were secured. The method of 
reduction is given on page 99. 

Two kinds of signals were used ; one consisted of a hemi- 
sphere of polished copper placed with its axis vertically over the 
centre of the station. The sun shining on this gave to the ob- 
server a bright point, but not in a line joining the centres of 
the stations observing and observed upon ; consequently a cor- 
rection for phase, as explained on page 144 had to be applied. 
The other form consisted of a board about two feet square, 
painted white with a black vertical stripe ten inches wide down 
the centre. This board was attached to an axis made to coin- 
cide with the centre of the station, so as to permit the board 
to be turned in a direction perpendicular to the line of sight as 
different stations were being occupied. 

The astronomic determinations were made at three stations 
with the greatest possible care ; while the reduction of the tri- 
angulation was a monument to the methods devised by Gauss 
for treating all auxiliary angles as aids in finding the most 
probable corrections to be applied to those angles absolutely 
needed in the computation. The amplitude of the arc was 
1° 3c/ 28' / .97. Using the two parts into which the arc was di- 
vided by Konigsberg, the difference between the terminal 
points taken as a whole, and the sum of the two parts was only 
0.973 toise, which is an evidence of the great accuracy attained 
in this work. The report of this triangulation was published 
in Gradmessung in Ostpreussen, und ihre Verbindung mit Preus- 
sischen und Russischen Dreiecksketten, Berlin, 1838; and while 
now nearly half a century has elapsed since its appearance, not 
only its influence is still felt, but the operations then for the 
first time described are now in use. 

There is not a geodesist of the present time who is not in- 
debted to this work for information as well as assistance, and 



22 GEODETIC OPERATIONS. 

as long as exact science receives attention men will turn to 
this fountain-head. My greatest inspiration comes from two 
sources — both perhaps sentimental, but none the less effica- 
cious. My copy of the above book was presented to Jacobi 
by Bessel, as shown by the latter's superscription. This is be- 
fore me in reality ; the other remains in memory as the cordial 
greetings and encouragement of Baeyer, with whom I worked 
in the Geodetic Institute. 

From 1843 to J 86i Sir A. Waugh, who succeeded Sir George 
Everest, added nearly 8000 miles to the Indian chains. After 
him came General Walker's administration, and during the 
following thirteen years he completed 5500 miles of triangle 
chains, occupied 55 azimuth stations, and determined 89 
latitudes. 

In this work the triangle sides are from 15 to 60 miles in 
length. In those cases where it was necessary to elevate the 
instrument masonry towers were erected, some as high as 50 
feet. Luminous signals were used — heliotropes by day, and 
Argand lamps at night. The amplitude of the greatest Indian 
arc is 23°4o/ 23". 54, but its exact value has 'been questioned, 
owino; to the uncertainties of the effect of local attractions in the 
neighborhood of the Himalayas upon the latitudes and azi- 
muths, as well as the negative attraction along the shore of the 
Indian Ocean as pointed out by Archdeacon Pratt. When 
the computed effects of these attractions are applied, there is 
still a discrepancy. 

A meridional arc of about 30 has been completed, but owing 
to the impracticability of ascertaining the difference of longi- 
tudes its amplitude is not accepted as sufficiently accurate to 
warrant its use in degree-determinations. 

The purpose of this great trigonometric survey was to fur- 
nish a basis for topographic maps ; consequently the chains of 
primary triangles are parallel at such a distance apart as to 
allow the intervening country to be easily covered with 



HISTORIC SKETCH. 23 

secondary triangles with the primaries for checks on each side 
of the chasm. There are 24 chains running north and south, 
and 7 east and west. 

Between 1847 anc * 185 1 the Russian chain was connected 
with the Austrian, having 12 sides in common; the greatest 
discrepancy being 0.101 toise, and the least 0.01 toise. 

About the same time the junction of the Lombardy and 
Swiss chains showed a difference of 0.31 and 0.34 metre. 

In 1848 the astronomer Maclear revised Lacaille's Good 
Hope arc, extending it to an amplitude of 3 degrees, from 
which he deduced for I degree, in latitude 35 43', 56,932.5 
toises. Comparing this with the French arc in approximately 
the same northern latitude, we find a difference of only 48 
toises in a degree. 

In 1 83 1 Borden devised a base-apparatus with which he 
measured a base and began a triangulation over the State of 
Massachusetts, making the commencement of geodetic work 
in the United States. Borden read his angles with a 12-inch 
theodolite, using the method of repetition. Latitudes were 
determined from circumpolar altitude observations at 24 
points. 

Recently many of his stations have been re-occupied, intro- 
ducing greater care in all features of the work and affording a 
check on Borden's results. Comparing the two sets of values 
for the geographical positions of the stations that are common, 
it appears that there is a systematic increase in the errors, being 
the greatest in the eastern part of the State, that being the 
furthest from the base-line. The average discrepancy in the 
linear determination is 1:11000, or somewhat less than 6 inches 
in a mile. 

The United States Coast Survey, organized in 1807, had 
primarily for its object the survey of the coast, but this ne- 
cessitated a carefully executed triangulation of long sides to 
check the short triangle sides whose terminal stations were 



24 GEODETIC OPERATIONS. 

sufficiently near one another for the coast topography and off- 
shore hydrography. It soon became apparent that but little, 
if any, additional care was needed to secure sufficient accuracy 
to make this trigonometric work a contribution to geodesy. 
By 1867 an arc of 3 23' was completed, extending from Farm- 
ington, Maine, to Nantucket, with two base-lines, seven latitude 
stations, and ten determinations of azimuth. 

Summing the six arcs into which the whole naturally divides 
itself, it was found that a degree in latitude 43 and longitude 
70°20 / was 111,096 metres, or 57,000.5 toises. 

By 1876 the Pamlico-Chesapeake arc of 4 31 '.5 was com- 
pleted, embracing in its chain of triangles six bases and fourteen 
astronomic stations. The latitude of each of these stations was 
computed from the one nearest the middle of the arc, and the 
difference between this and the observed values, called station- 
error, attributed to local deflection. This in no case exceeded 
3-! seconds; and in general it was in accord with a uniform 
law disclosed by the geology of the country over which the arc 
extends. 

From an elaborate discussion of the sources of error in this 
arc, Mr. Schott concludes that the probable error in its length 
is not in excess of 3J metres. The length of a degree in lati- 
tude 37 16' and longitude ?6 C 08' is 56,999.9 toises. 

The triangulation is being continued southward, and in a 
very short time it is hoped that the entire possible arc of 22° 
will be reduced and the results announced. An arc of parallel 
is also under way, keeping close to the 39th. 

Of this great arc of 49 about three fourths is completed. 
This is the longest arc that can anywhere be measured under 
the auspices of a single country. Consequently, considering 
the great advantage to be derived from perfect harmony of 
methods, it is no wonder that scientists in all parts of the 
world are anxiously awaiting the completion of this important 
work. Also, when done, it will be well done. The high stand- 



HISTORIC SKETCH. 2$ 

ard of excellence introduced into this service at its beginning 
makes the first results comparable with the most recent. 

In 1857 Struve advocated the project of connecting the 
triangulations of Russia, Prussia, Belgium, and England, giving 
an arc of 69 along the 52d parallel. Bessel had already made 
the Prussian-Russian connection, and in 1861 England and Bel- 
gium joined with tolerable success, finding in their common 
lines discrepancies amounting to an inch in a mile. 

The Prussian and Belgium chains are not yet satisfactorily 
united ; neither are the longitudes determined. 

While a topographic map of Italy was begun in 1815, no 
special interest was taken in geodesy until 1861, except in 
rendering some slight assistance in that part of the French and 
Austrian triangulation that overlapped. In this year Italy re- 
sponded to the suggestion of Baeyer, adopted by the Prussian 
Government, to form an association of the European powers to 
measure a meridional and a parallel arc. 

The Italian Commission was formed in 1865, and at once 
elaborated plans for future work. It was decided to have six 
chains of triangles, and for every twenty or twenty-five a care- 
fully measured base ; also to connect Sicily with Africa ; direc- 
tion-theodolites of 10- and 12-inch circles to be used. The base- 
apparatus with which the first three bases were measured was 
of the Bessel pattern. The base of Undine was measured with 
the Austrian, and the next two with a Bessel equipped with 
reading-microscopes for reading the divisions on the glass 
wedges. 

The numerous observatories are connected with the trigono- 
metric stations, and one or two are to be erected in the merid- 
ian of the arc to determine its deflection. 

The geodetic work in Spain began with the measurement of 
the Madridejos base in 1858. The apparatus used in this work 
was specially designed for it, and the precision introduced 
into the measurement of the base, as well as in the depend- 



26 GEODETIC OPERATIONS. 

ing triangulation, has given to the Spanish work great confi- 
dence. 

This is especially fortunate, as it will form an important link 
in the chain extending from the north of Scotland into Africa, 
and in the oblique chain from Lapland to the same point. In 
addition to the central base first measured, three others were 
found necessary to check the system. 

The general plan resembles that pursued in the India Survey 
in having parallel chains at such a distance from one another 
that the intervening country can be readily filled in with sec- 
ondary triangles for the topographic purposes. 

There are three of these meridional chains with amplitudes 
of about six, seven, and seven and a half degrees, and an arc of 
parallel of twelve degrees. 

Likewise the Swedish coast-triangulation was begun in 1758 
for the purpose of checking the coast-charts, and in 18 12 an- 
other triangulation embracing fifty stations and five base-lines, 
measured with wooden rods, was started for a similar end. 
However, it was not until the announcement of Bessel's results 
that Sweden took an active interest in accurate work. 

In 1839 the Alvaren base was measured with Bessel's appara- 
tus, and again in the following year with the same bars, giving 
a difference of 0.0145 metre in the two results. 

So far the work was purely cartographic, and it was the in- 
fluence of Baeyer that caused a partial transformation in the 
methods, making them conformable to the system of the Per- 
manent Commission for European Degree-measurements. 

Three bases have been measured with a modified Struve ap- 
paratus, giving excellent results; in one instance the difference 
between the two measurements being only 0.0029 metre, and 
twenty-nine stations occupied, using Reichenbach and Repsold 
theodolites. 

Under the auspices of this commission the following coun- 
tries are prosecuting geodetic work : Austria, Bavaria, Belgium, 



HISTORIC SKETCH. 2J 

France, Hesse, Holland, Italy, Portugal, Prussia, Russia, Sax- 
ony, Spain, Switzerland, and Wiirtemberg. 



LITERATURE OF THE HISTORY OF GEODESY. 

Verhandlungen der allgemeinen Conferenz der Europaischen 
Gradmessung. 

Roberts, Figure of the Earth, Van Nostrand's Engineering 
Magazine, vol. 32, pp. 228-242. 

Comstock, Notes on European Surveys. 

Baeyer, Ueber die Grosse und Figur der Erde. 

Posch, Geschichte und System der Breitengradmessungen. 

Merriman, Figure of the Earth. 

Baily, Histoire de l'Astronomie. 

Wolf, Geschichte der Vermessungen in der Schweiz. 

Clarke, Geodesy. 

Westphal, Basisapparate und Basismessungen. 

Klein, Zweck und Aufgabe der Europaischen Gradmessung. 



28 GEODETIC OPERATIONS. 



CHAPTER II. 

INSTRUMENTS AND METHODS OF OBSERVATION. 

THE perfection of an instrument is the result of corrected 
defects, and in the development of geodesy or degree-meas- 
urements improved methods were closely followed by better 
instruments. So that while discussing the progressive steps 
of one, the other cannot be wholly neglected. 

For the uncultured peoples, distances can be given with suf- 
ficient accuracy as so many days' journey, and nothing but the 
necessity to carry on record some measured magnitude would 
call for a unit that could be readily attained. The first such 
unit of which there is any authentic information is the Chaldean 
mile, which was equal to 4000 steps of a camel ; the next was 
the Olympian race-course, giving to the Greeks their unit — 
the stadium. The rods with which the Arabians measured the 
two degrees already mentioned — known as the black ell — 
have been lost, and not even their equivalent length known. 

Fernel, in using the wagon-wheel for a measuring unit, found 
it quite constant in length and of a kind easily applied, — advan- 
tages that are appreciated to this day by topographers, who 
frequently measure meander lines by having a cyclometer at- 
tached to a wheel of a vehicle. 

When Snellius devised the method of triangulation there 
were needed two forms of instruments — one for linear measure- 
ments, and another for angle-determinations. At this time 
angles were measured with a quadrant to which sights were 
attached ; a rectangle with an alidade and sights pivoted to one 
of the longer sides, the other being divided into degrees ; a 



INSTRUMENTS AND METHODS OF OBSERVATION. 29 

square with the alidade in one corner and all four sides gradu- 
ated ; a compass with sights ; a semicircle with alidade or 
compass at the centre. Also for navigators there was the as- 
trolabe, an instrument devised by Hipparchus for measuring 
the altitude of the sun or a star. 

Defects in graduation were early detected, and efforts to 
avoid them made by increasing the radius of the sector, the 
smallest used by the first astronomers being of 6 and 7 feet 
radius ; and it is said that a pupil of Tycho Brahe constructed 
a sector of 14 feet radius; while Humboldt says the Arabian 
astronomers occasionally employed quadrants of 180 feet ra- 
dius. In the case of large circles, or parts of circles, the divi- 
sions that could be distinguished would be so numerous as to 
render the labor of dividing very great, and the intermediate 
approximation uncertain. 

Nunez, a Portuguese, in 1542 devised a means of estimating 
a value smaller than the unit of division. He had about his 
quadrant several concentric circular arcs, each having one divi- 
sion less than the next outer, so that the difference between an 
outer and an inner division was one divided by the number of 
parts into which the outer was divided. This differs from our 
present vernier, first used by Petrus Vernierus in 1631, in which 
the auxiliary arc is short and is carried around with the zero- 
point. 

A great impetus was given to applied mathematics by the 
construction of logarithmic tables according to the formulae of 
Napier (1550-1617), and Briggs(i 556-1630), especially in facili- 
tating trigonometric computations, which had now become the 
basis of degree-measurements. 

The first person to use an entire circle instead of a part was 
Roemer in 1672, who deserves our thanks for having invented 
the transit also. Auzout in 1666 made the first micrometer, 
and Picard was the first to apply it, and a telescope with cross- 
wires, to an angle-reading instrument. The results obtained 



30 GEODETIC OPERATIONS. 

with this instrument were so satisfactory that Cassini used it 
in his great triangulation begun eleven years later. The angles 
in Peru were measured with quadrants of 21, 24, 30 and 36 
inches radius, each provided with one micrometer. These 
gave very fair results — the maximum error in closure of a tri- 
angle being 12 seconds, spherical excess not considered. This 
would give an error of one unit in 5000 in the length of a de- 
pending line — a value ten times better than any obtained dur- 
ing the preceding century. 

With such close reading of angles the discrepancies between 
measured and computed lines were quite naturally attributed 
to the unit of measure, the method of its use, or its comparison 
with a standard. As early as the Peruvian work the uncer- 
tainty in the varying length of wooden rods because of damp- 
ness, and of metal rods on account of heat, was appreciated ; 
and in the measurement of these Jjases an approximate average 
of 1 3 R. was assumed for the mean temperature. This hap- 
pened to be the temperature at which the field standards had 
been compared with the copy before leaving Paris, hence the 
reason for legalizing this temperature for that at which the 
toise of Peru is a standard. 

In 1752 Mayer announced the advantages to be derived 
from repeating angles, and a repeating-circle was constructed 
upon this principle by Borda in 1785, for the connection of the 
French and English work. The first dividing engine was made 
by Ramsden in 1763, and a second improved one in 1773, 
which did such good work that his circles soon became deserv- 
edly famous. In 1783 this maker furnished an instrument to 
the English party engaged upon the work just mentioned, 
this was the first to be called theodolite. It had a circle three 
feet in diameter, divided into ten-minute spaces, read by two 
reading micrometer microscopes. One turn of the micrometer- 
screw was equal to one minute, and the head was divided into 
sixty parts, so that a direct reading to a single second could 



INSTRUMENTS AND METHODS OF OBSERVATION, 3 1 

be made, and to a decimal by approximation. It was also 
provided with a vertical circle of 10.5 inches diameter, read by 
two micrometers to three seconds. The success attained in 
the use of this instrument, giving a maximum error of closure 
of three seconds, was regarded as truly phenomenal. 

Reichenbach began the manufacture of instruments, in 
Munich, in 1804, of such a high grade of workmanship that it 
was soon considered unnecessary to send to Paris or London in 
order to secure the best. He fortunately furnished Struve with 
a theodolite, putting a good instrument in the hands of one of 
the most skilful observers who has ever lived, which contrib- 
uted no little to his reputation. His circles were almost wholly 
repeaters, a class of instruments exclusively used on the Con- 
tinent, but not at all in England. 

Littrow, at the Observatory of Vienna, was the first to aban- 
don the method of repetition, in 1819; and Struve, in 1822, was 
the next to follow. 

The inconvenience attending the use of large circles was 
very great, besides the irregularities produced from flexure on 
account of unequal distribution of supports. This led to the 
attempt to make a smaller circle with good graduation, and 
reading-microscopes. This end was achieved by Repsold, who 
made a ten-inch theodolite for Schumacher in 1839, with which 
it was definitely demonstrated that as good results could be se- 
cured with a ten or a twelve inch instrument as with a larger one, 
and with less expenditure of time and labor, not considering 
the difference in the first cost. So that now we find the effort 
heretofore spent in constructing enormous circles given to per- 
fecting the graduation, and, while using the instrument, to pro- 
tect the circle from sudden or unequal changes of temperature. 
Mr. Saegmuller's principle of bisection in dividing a circle 
keeps the errors of graduation within small limits, and the new 
dividing engines leave but little to be desired in the construc- 
tion of theodolites. 



32 GEODETIC OPERATIONS. 

In England and India eighteen-inch circles are now used 
in place of those of twice that size formerly employed. 
Struve had a thirteen-inch theodolite. In the U. S. Coast and 
Geodetic Survey the large instruments have given way to those 
of twelve inches. In Spain twelve- and fourteen-inch circles are 
found to be the best, while the excellent work of the U. S. Lake 
Survey was done with theodolites having circles of twenty and 
fourteen inches in diameter — the latter having the preference. 

To describe the various forms of theodolites now in use 
would necessitate a number of illustrations, and in the end be 
tedious and unprofitable; the same general features being com- 
mon to all, they only will be referred to. The end sought in 
the construction of theodolites is to get an instrument with 
parts sufficiently light to insure requisite stability, with circles 
large enough to allow close readings, with the telescopic axis 
concentric with the circle, a reliable means for subdividing the 
divisions on the circle, and a circle so graduated as to be free 
from errors, or to have them according to a law readily dis- 
tinguished and easily allowed for. While every one concedes 
that the foregoing requisites are imperative, in respect to some 
there is a great difference of opinion as to when they are at- 
tained. 

The illustration appended shows an eight- to twelve-inch 
theodolite of the form suggested by the experience of the skilled 
officers of the U. S. Coast and Geodetic Survey. In its construc- 
tion hard metal is employed, and as few parts used as possible. 
The frame is made of hollow or ribbed pieces in that shape 
that gives the greatest strength for the material. The bearings 
are conical ; clamps of a kind that avoid travelling motion ; the 
circle is solid, and of a conical shape to prevent flexure. The 
focal distance is diminished so as to admit of reversal of tele- 
scope without removing it from its supports, and the optical 
power is increased to insure precision in bisecting a signal. 
They are made as nearly symmetrical as possible, and when 



INSTRUMENTS AND METHODS OF OBSERVATION 33 

there is no counterpoise provided, one of the proper weight is 
put in place. They are furnished with three foot-screws for 
levelling, resting in grooves converging towards the centre. 
Sometimes a circular level is set in the lowest part of the 
branching supports, and in other cases a single tubular level is 
made use of. The optical axis is marked by having in the 
principal focus spider-lines called a reticule, or a piece of very 
thin glass on which fine lines are etched. The arrangement of 
the lines is various, the forms depicted in the annexed cut be- 
ing the ones most frequently found. 




The instrument shown in Fig. 1 is one of directions in which 
the circle is shifted for new positions. With a repeater the 
only difference is the addition of a slow-motion screw to move 
the entire instrument in accordance with the method of repe- 
tition as explained on page 98. 

The adjustments of a theodolite must be carefully attended 
to and frequently tested. They may be described in general 
as follow : 

To Adjust the Levels. — When the tripod or stand is placed in 
a stable condition and the instrument mounted, bring it into a 
level position, as indicated by the level, by turning the foot- 
screws. Turn the instrument 180 degrees, correct any defect, 
— one half by means of the screws attached to the level, and 
the rest by the foot-screws. Place the instrument in its first 
position, repeat the corrections as before until no deviation is 
noticed when the circle is turned. If there is a second level, it 
is to be adjusted in the same manner. 
3 



34 



GEODETIC OPERATIONS. 




Fig. x. 



INSTRUMENTS AND METHODS OF OBSERVATION. 3$ 

To Adjust the Spider-lines of the Telescope.- (1) Place the 
threads in the focus of the eye-piece, point to a suspended 
plumb-line when the air is still, and see if the vertical thread 
coincides with the plumb-line. If there is any deflection, loosen 
the four screws holding the diaphragm and move it gently till 
there is a coincidence, then tighten the screws and verify. (2) 
If the level is correct, place the circle in a horizontal position 
and sight to some clearly defined object ; move the instrument 
sideways by means of the tangent screw and notice if the hori- 
zontal thread traverses the point throughout its entire length, 
if not, correct as in the above case. 

To Adjust the Line of Collimation of the Telescope.— -When 
the horizontal axis of the telescope can be reversed, point the 
instrument to some clearly defined object, then reverse the 
telescope and see if the pointing is good. If not, half the dif- 
ference is to be corrected in the pointing and the other half 
by moving the entire diaphragm to the right or left, as the case 
may be. Continue this course until the pointing remains per- 
fect after reversal. If the instrument does not admit of this 
reversal, it must be turned in its Y's ; and if the reading is more 
or less than 180 degrees from the first reading, correct as be- 
fore, until there is just 1 80 degrees between the readings before 
and after reversal. m 

The horizontally of the axis of the telescope is tested by 
placing on the axis a portable level that is in good adjustment. 
If a defect is apparent, it must be corrected entirely by raising 
or lowering the movable end. 

After completing these adjustments, it is well to repeat the 
tests to see if any have been disturbed while the other ad- 
justments were in progress. When large instruments with 
reading-microscopes are used, the corrections for runs and 
eccentricity must be determined. The former can be readily 
ascertained as follows: Turn the micrometer in the direction 
of the increasing numbers on its head till the movable cross- 



$6 GEODETIC OPERATIONS. 

wire bisects the first five-minute space ; call the reading a. 
Reverse the motion and continue to the preceding five-minute 
space ; call this b. Suppose 

a = 45° 40' + A' 46"4, b = 45° A°' + 4' 44"2, 

r = a - b = + 2".2, ** = ( -^±_^ = 4 ' 45 ". 3 . 

Since the five-minute space contains 300 seconds, the correction 
to # = r . tf -f- 300 = — 2" ,\ ; correction to £ = ;-(£ — 300) 
-f. 300 = -\- " - 1 1 ; correction to m = J(# -f- b — 300)^ -=- 300 
= — o".88. The corrected reading is therefore, 

45°44 / 45 // -3 - .88 =45° 44' 44^.42. 

Occasionally the average error of runs is determined and a 
table computed from the formula just given for a -\- b from 5 




Fig. 2. 



to 10 seconds. But in very accurate work the correction for 
runs is made for each reading by recording the two micrometer- 
readings just mentioned for each pointing. They are recorded 
as forward and backward, as seen on page 101. 

The eccentricity is owing to the centre of the axis carrying 
the telescope not coinciding with the centre of the graduated 
circle. As each point on the plate carrying the telescope must 



INSTRUMENTS AND METHODS OF OBSERVATION. 37 

return to its former position after each complete revolution, 
there must be a point at which there is a maximum deflection 
as well as a point at which there is no deflection, and at the 
same time the intermediate positions have eccentric errors be- 
tween these limits ; therefore it is necessary to examine the 
whole circle. This can be done in connection with an exami- 
nation of the two verniers. The difference in the reading of the 
two verniers may, however, be due to other causes : the con- 
stant angular distance between them may be more or less than 
180 degrees, or it may be owing to errors of graduation, or 
errors of reading, or to the eccentricity referred to. 

Let c be the centre of the limb, 
m that of the telescope, 
6 = angle amb, 
6' = angle deb, 
E = the difference, or error, 
e = cm = the linear eccentricity, 
go z= dem, 

r = radius of the circle, 
d = cdm, 
b = cbm, 
dm = 6 + b = 6'+d; therefore, £=6-6'=d-b. 

As cm is never very large, we can put mb = r : in the tri- 
angle cdm, we have sin d = — sin go, and in the triangle ban, 
we have 

€ 6 

sin b = — sin bem = — sin (go — 6'). 
r r v J 



Also, since d and b are small, we can write for sin b, b. sin 1", 
and for sin d, d .sin 1", so that we have, 



38 GEODETIC OPERATIONS, 

g 

E = d — b = : ^sin g? — sin (o> — 0'Y|. 

r . sin i //L v JJ 

By expanding sin (g? — #'), and putting for the entire angles 
their values in terms of the half-angles, we find, 



E = 



2e 



r . sin 



p 7 [sini6>\cos(G*-i<90]. 



This expression is made up of two factors, and becomes o 
when either factor becomes o, as e = o, or cos (go — %d') = o, 
that is, when go — \B' = 90 , or 6' = 2go — 180 . 

Therefore when the points are 180 apart the errors of ec- 
centricity are eliminated. Likewise E is a maximum when 
cos (go — %d') = -j- 1, that is, when go — -|#' = o, or 2go = 6'. 

In accord with the principle that errors of eccentricity are 
avoided when the angle is read from two points 180 apart, 
circles are provided with two verniers that distance from each 
other. Instead of verniers, however, we may have two micro- 
scopes. 

The practical difficulty of placing the zero-points just 180 
apart makes it necessary to examine each circle to see what 
the angular distance between them is. This is best accom- 
plished by setting one vernier, say A, on each 10° mark, and 
reading and recording vernier B. If a represent the amount 
by which the angular distance differs from 180 , and b the effect 
of eccentricity on this distance, we will have B — A = 180 
-J- a -f- b, and when the verniers change places b will have a 
contrary effect, so that B —A = 180 -f- a — b ; therefore if we 
take the mean of the differences B — A for positions that are 
just 180 apart, we will have the angular distance unaffected by 
eccentricity. We so arrange our readings as to have on the 
same line those that are 180 apart. We also place under B 
— A the first difference, and on the same line the second dif- 
ference, the mean will be the average of the two, or 180 + a, 



INSTRUMENTS AND METHODS OF OBSERVATION. 39 

and the average of these means will be the mean distance be- 
tween the verniers. 



First. 


Second. 


B - A. 


A. 


B. 


A. 


B. 


I St. 


2d. 


Mean. 


o° 00' 00" 


180 00' 05" 


180 oo' 00" 


o° 00' 00" 


+ 5" 





+ 2". 5 


IO 


10 


190 


05 


+ 10 


+ 5 


+ 7 -5 


20 


05 


200 


OO 


+ 5 





+ 2 .5 


30 


10 


210 


05 


+ 10 


+ 5 


+ 7 .5 


40 


55 


220 


OO 


- 5 





-2.5 


50 


00 


230 


05 





+ 5 


+ 2 .5 


60 


05 


240 


IO 


+ 5 


+10 


+ 7 -5 


70 


05 


250 


05 


+ 5 


+ 5 


+ 5 


80 


10 


260 


OO 


+ 10 





+ 5 


90 


05 


270 


IO 


+ 5 


+10 


+ 7 .5 


IOO 


00 


280 


55 





- 5 


-2.5 


no 


55 


290 


00 


- 5 





-2.5 


120 


55 


300 


05 


- 5 


+ 5 





130 


05 


310 


05 




- 5 


+ 5 


+ 5 


140 


05 


320 


55 




- 5 


- 5 





150 


05 


330 


00 




- 5 





+ 2 .5 


160 


05 


340 


00 




- 5 


+2 .5 


170 


05 


350 


05 




- 5 


+ 5+5 



Therefore the angular distance = 180 -j- 3 ;/ -i- Mean = 3 ;, .i. 

Now, knowing the angular distance between the two verniers, 
the difference between it and the mean of B — A will be the 
errors of eccentricity and graduation, or b -\- g. 

Angle dcA = m-\-A> therefore A = 
dcA — m. If we call dthe reading on the 
limb which is on the line of no eccentricity, 
that is on the line drawn through the 
centre of motion and centre of graduation, 
and n any angle read by the verniers, 
then 11 — d will be the angle between the 
vernier and line of no eccentricity, or dcA. 
In the triangle Acm, sin Acm : sin A :: 
Am : cm, but sin Acm = sin dcA = sin {it 
nearly, making these substitutions: 




Fig. 3. 

d), and Am = r, 



40 GEODETIC OPERATIONS. 

• / _7\ • a . . *. sin (n—d) 
sin (n — a) : sm A :: r : e, or sin ^4 = -. 

A being small, we can put for sin A, A . sin i", and the angu- 
lar value for ^ to radius r, e . sin \" ; then write for A in 
seconds, ^4 = e. sin (# — d), and for the two verniers, b = 
2^. sin {n—d). A reading b r at i8o° from the former will 
have the same error, but with an opposite sign, b' = — 2e 
. sin (it — d). If we tabulate the differences between the 
mean in our first table and the various readings for B — A, 
placing on the same line those that differ by i8o° from one an- 
other, they should be equal with opposite signs were it not for 
errors of graduation ; let these differences be D and D ', then 
b+g=D,zn&b'+g=D' y 

2e sin (n — d)-\-g = D 
— 2e sin (it — d) -\- g = D' 

2g=D + D\ g = i(D + B'). 

Subtracting, 

4e sin (n — d) = D — D\ 2e sin (« - d) = i(D - D') = b, 

or a value for b freed from errors of graduation. This will give 
l8 equations involving e and n. 
Placing S = %(D — D r ), we have ; 

S x = 2es'm(o°—d)= 2^(sino° cosd— cos o° sin d)= — 2esind; 
S 2 = 2e sin ( io° — d) — 2^(sin io° cosd— io° sin*/); 



d 1% = 2^ sin (170 — d) = 2e(sln 170 cos d — cos 170 sin d). 



INSTRUMENTS AND METHODS OF 0BSERVA1I0N. 41 

Professor Hilgard's method for solving these equations with 
respect to 2e cos d and 2e sin d, by least squares, is to multiply 
each equation through by cos ft, and sum the resulting equa- 
tions ; then each through by sin n, and sum the results : this 
will give us two normal equations of this form ; after factoring 
2e cos d, and — 2e sin d, 

[S x sin o° + 6 % sin io° . . . 6 l% sin 170 ] 

= 2e cos d [sin 2 o° + sin 2 io . . . sin 2 170 ] 

— 2e cos d [sin o° cos o°+sin io° cos io°-f-. . . sin 170 cos 170 ]; 

\? x cos o° + 6 9 cos io° . . . <? 1B cos 170 ] 

= 2e cos ^[cos o° sin o° . . . cos 170 sin 170 ] 
— 2e sin ^[cos 2 o° -f- cos 2 io° . . . cos 2 170 ] 

sin o° cos 0° = O, also for sin io° cos io° we can put -Jsin 20 

and so on with all the products of sines times cosines ; and we 

find that this will give us pairs of angles that make up 360 , 

whose sines are equal but with opposite algebraic signs, so the 

products reduce to zero. Again, we can arrange the second 

powers so that all angles above 90 can be written 90 + n 5 

sin 2 (90 -f- n) = cos 2 ft, this added to sin 2 ^=1, for example; 

sin 2 o° = o, sin 2 10° + sin 2 ioo° = sin 2 io° + sin 2 (90 s -f- io°) = 

sin 2 io° + cos2 l 0° = I. 

This will give us half as many unities as we have terms less 

N 
two for the pairs, and sin 2 90 = 1 gives us 9 = — . The nor- 

mal equations will then reduce to 

2{d sin n) = Ne cos d, 
2(3 cos n) = — Ne sin d ; 

.. . . ^(tfcosTz) 

by division, ^..^ ; = — tan d. 

J 2(d sin ft) 



42 



GEODETIC OPERATIONS. 



n. 


First 


Second 


ISt-2d =8 sJl 

2 


1 «. CO 


s «. 


(8 sin 


n. 


1 

h cos n. 


o° 


+ 2.1 


- 2.9 


+ 2.5 O 


00 I 


OO 


O" 


..OO 




-2". 50 


IO 


+ 7.1 


+ 2.1 


+ 2.5 


17 


98 


+ 


•43 




-2 .45 


20 


+ 2.1 


- 2.9 


+ 2.5 


34 


94 


+ 


.85 




-2 -35 


30 


+ 7-1 


+ 2.1 


+ 2.5 


50 


87 


+ 1 


.25 




-2 .17 


40 


- 7-9 


- 2.9 


— 2.5 


64 


76 


— I 


.60 


— I .90 


50 


- 2.9 


+ 2.1 


- 2.5 


76 


64 


— I 


.90 


— I .60 


60 




- 2.1 


+ 7.1 


- 2.5 


87 


50 


— 2 


.17 


- I .25 


70 




- 2.1 


+ 2.1 


O 


94 


34 


O 


.OO 


O .OO 


80 




- 7-1 


- 2.9 


+ 5.0 


98 


17 


+ 4 


.90 


+ .85 


90 




- 2.1 


+ 7-1 


- 2.5 I 


00 


00 


— 2 


.50 


O .OO 


IOO 


- 2.9 


- 7-9 


+ 2-5 


98 - 


17 


+ 2 


•45 


- -43 


no 


- 7.9 


- 2.9 


~ 2.5 


94 - 


34 


— 2 


•35 


+ .85 


120 


- 7-9 


+ 2.1 


- 5.0 


87 - 


50 


- 4 


•35 


+ 2 .50 


130 


+ 2.1 


+ 2.1 


O 


76 - 


64 





.00 


.00 


140 


+ 2.1 


- 7-9 


+ 5-o 


64 - 


76 


+ 3 


.20 


— 3 .80 


150 


+ 2.1 


- 2.9 


+ 2.5 


5o - 


87 


+ 1 


.25 


- 2 .17 


160 


+ 2.1 


- 2.9 


+ 2.5 


34 - 


94 


+ 


.85 


- 2 .35 


170 


+ 2.1 


+ 2.1 





17 ~ 

2(8 sin 


98 
n) = 





.00 


.00 


+ 


•31 












.S^ cos t 


= 






+ .17 



tan d = 



^ = 



0.17 



0.3 



= tani5i° 15' 40"; 
0.17 



18 sin 151 15' 40""" ° '° 2 ' 



The line of no eccentricity is that passing through 15 1° 15' 
40"; the sign of e being minus, we know that the centre of mo- 
tion is in the opposite direction from the centre of graduation 
towards the reading d. In this case it is too small to be con- 
sidered. To determine the error of graduation, we compute 
the values of 2e sin (it — d) = b; subtracting these results 
from those in the last table marked b -^ g'm the first column, 
we will have£\ It is necessary to compute b for every io° 
space, only up to 180 , since b has the same value for 180 
+ n that it has for n, with the opposite sign, then subtract 
these values from the second b -\- g. 



INSTRUMENTS AND METHODS OF OBSERVATION. 43 



«. 

o° 


n - 


-d. 
151° 


ie sin 


(« - d,. 


n. 


sr- 


». 


.r- 


"+" 


O.OI7 


— 0° 


+ 2" 


.083 


180 


— 2.883 


TO 


— 


141 


+ 


.025 


IO 


+ 7 


•075 


190 


+ 2.125 


20 


— 


131 


+ 


.030 


20 


+ 2 


.070 


200 


— 2.870 


30 


— 


121 


+ 


.034 


30 


+ 7 


.066 


210 


+ 2.134 


40 


— 


III 


+ 


.037 


40 


- 7 


•937 


220 


— 2.863 


SO 


— 


ior 


+ 


.038 


50 


— 2 


•933 


230 


+ 2.13S 


60 


— 


9 1 


+ 


.040 


60 


+ 2 


.060 


240 


+ 7-140 


70 


— 


Si 


+ 


.038 


70 


+ 2 


.062 


250 


+ 2. 138 


SO 


— 


7i 


+ 


•037 


80 


+ 7 


.063 


260 


— 2.863 


90 


— 


61 


+ 


.034 


90 


+ 2 


.066 


270 


+ 7.134 


IOO 


— 


5t 


+ 


.030 


IOO 


— 2 


•930 


2SO 


- 7-870 


no 


— 


4i 


+ 


.026 


no 


- 7 


.926 


29O 


- 2.874 


120 


— 


3i 


+ 


.020 


I20 


— 7 


.920 


300 


-f- 2.120 


130 


— 


21 


+ 


.014 


I30 


+ 2 


.086 


3IO 


+ 2. 114 


140 


— 


11 


+ 


.007 


I40 


+ 2 


093 


320 


- 7 893 


150 


— 


01 


+ 


• OOO 


I50 


+ 2 


.100 


330 


— 2 900 


160 


+ 


09 




.006 


l6o 


+ 2 


. 106 


340 


— 2.906 


170 


+ 


19 


— 


.013 


I70 


+ 2 


•113 


350 


+ 2.087 



The sum of the squares of the 36 values for g give 714.7617, 
therefore the probable error in any one is a ' ^ — L = 

± 4 // .6 ; this divided by the square root of two gives the prob- 
able error of the reading of one vernier, owing to errors of grad- 
uation and accidental errors of reading = ± 3 // .2. 

If an angle is the mean of five repetitions, the probable error 
of the average will be one fifth of 3'' ' .2 = ± o / '.64 < 

If the effect of eccentricity be considerable, the correction to 
each angle should be computed by the equation b = 2e sin 
{n — d). The probable error in graduation and reading is 
used only in computing the probable error in a chain of tri- 
angles, as will be seen later. If the instrument has two read- 
ing-microscopes the procedure is essentially the same, but dif- 
fers slightly when there are three. In this case every 5 or io° 
space can be examined and the three microscopes read ; as be- 
fore, we shall call the reading of the zero-point n, and the 
microscopes A, B, and C. 



44 



GEODETIC OPERATIONS. 





n. 


n -f- 120*. 


n -j- 240*. 




o° oo' oo" 

I20 OO OI 

240 00 02 


+ OI" 
OO 

- 04 


+ 02" 
— OI 
OO 


Sum 
One third 


+ 03 

— 02.6 

— 00.9 


- 03 
+ 03.3 

-f- OI.I 


+ 01 

— 0.6 

— 0.2 



= -f- i» average = + 0.3. 



The first line gives the readings when the zero-point is n, 
the order of the microscopes is A, B, and C; in the next, zero 
is at n -f- 120 , and the order is C, A, and B ; in the third, zero 
is at n-\- 240 , and the order is B, C, and A. The fourth line 
contains the sums, and the continuation the average ; and by 
subtracting the sums from this average we have the fifth line 
containing three times the errors of trisection at this point. 

Eccentricity is first determined : " Suppose a m /3 n , and y n be 
the observed errors of trisection corresponding to 11, n-\- 120 , 
and n + 240 , also [a n cos »], [fi„ cos(n -f- 120 )], [a n sin ri\ . . . 
etc., be the sums of all the a n cos n, a n sin n, etc., then d, the 
line of no eccentricity, 

\a n cos n\ -f [/3 n cos (n + 120 )] + [y„ cos {it + 240 )] 
" [a n sin »] + [/4 sin {n + 120 )] + [y„ sin (n -j- 240 )] 

also e" 

__ [ a * sin n\ + [/?« s^ («+ 120°)] + [>„ sin (« -f 240 )] 

f iV sin d 



where N — number of trisections. 

" The correction for eccentricity is b = e sin (n — d) t then if 
&n , Pn, Yn' == errors of trisections freed from errors of eccen- 
tricity, we will have : 



INSTRUMENTS AND METHODS OF OBSERVATION. 45 

a H ' = a n — e sin (n — d) ; 

p n ' = fi u - e sin (n + 120 - d) ; 

r«' = r« — * sin (» + 2 4o° — ^)- 

Knowing cc n \ /3 M ', y n \ the residuals are squared, and the prob- 
able error of graduation and reading found as in the preceding 
case." 

Considering that the determination of latitude, longitude, 
and azimuth forms a part of practical astronomy, the only in- 
struments that remain to be described are the base-apparatus 
and heliotrope. The former is referred to in the chapter on 
base-measuring, and the latter can be dismissed with a few 
words. 

The first heliotrope was used by Gauss in 1820. It was 
somewhat complicated, consisting of a mirror attached to the 
objective end of a small telescope. This mirror had a narrow 
middle-section at right angles to the rest of it ; this was in- 
tended to reflect light into the tube, while the remainder re- 
flected the sun's rays upon the object towards which the tele- 
scope was pointed. Bessel devised a much simpler form that is 
still in use in Prussia. It has a small mirror, with two motions, 
fastened to one end of a narrow strip of board, while at the 
other end there is a short tube whose height above the board 
is the same as the axis of the mirror. In this tube cross-wires 
are stretched, and a shutter can be dropped over the end op- 
posite the mirror. To use it, one fastens the screw that is at- 
tached to one end in a suitable support and then by means 
of a levelling screw at the other end, raises or lowers that 
end until the centre of the mirror, the cross-wires and the 
object towards which the light is to be reflected are in line. 
The mirror is then turned so that the shadow of the cross- 
wires falls upon their counterpart that is marked on the shutter 
when the light can be seen at the desired point. Perhaps the 
most convenient of all is the heliotrope that finds employment 



46 GEODETIC OPERATIONS. 

in the U. S. Coast and Geodetic Survey. It can be seen in 
Fig. 4. First of all, there is a low-power telescope provided 
with a screw for attachment to a tree or signal. On one end 
of the tube is a fixed ring of convenient diameter, say one 
and a half inches, while at the other end is a mirror of two 
inches in diameter, and at an intermediate point, nearer the 
mirror, is another ring of the same height and size as the other, 
but clamped to the tube, admitting of a motion around it. 

To describe its use we will suppose it in adjustment. After 
having screwed it to a post, the telescope is turned until the 
cross-wires approximately coincide with the point to which the 
light is to be shown ; then turn the mirror so that the shadow 
of the nearer ring exactly coincides with the other ring. Then 
as the earth revolving places the sun in a different relative 
position, it will be necessary to continually move the glass in 
order to keep the shadow of the back ring on the front one. 
If the sun is behind the heliotrope an additional mirror will be 
needed to throw the light upon the glass. 

To effect the adjustment, it is necessary to have in the con- 
struction the centres of the rings and the mirror at the same 
distance from the optical axis of the telescope. Bisect some 
clearly defined point, then sight over the tops of the mirror 
and rings, turning the movable one until they are all in line 
with the object bisected by the telescope. Owing to the large 
diameter of the sun, a slight error in adjusting will not affect 
the successful use of this kind of a heliotrope. . 

When the observed and observing stations are within twenty 
miles of one another, the light spot may be too large to be 
easily bisected ; then it is best to place between the glass and 
rings a colored glass (orange is preferable), so as to reduce the 
light as seen to a mere spot. A code of signals can be adopted 
and messages exchanged between observer and heliotroper, 
such as " Correct your pointing," " Stop for the day," " Set 
on new station," " Too much light," " Not enough light," by 



INSTRUMENTS AND METHODS OF OBSERVATION. /fl 

cutting off the light with a hat or small screen ; a long stoppage 
standing for a dash, and a short one for a dot, when the words 
can be spelled out by the Morse code. 

The maximum distance at which a heliotropic signal can be 
seen depends upon the condition of the atmosphere. Perhaps 
the greatest was on the " Davidson quadrilateral," where a 
light was seen at a station 192 miles away. 

A very convenient form of heliotrope, especially for recon- 
noissance, is one invented by Steinheil, and known by his name. 
It differs from all others in having only one mirror and no 




Fig. 



rings, making it so simple in use and adjustment as to form 
a valuable instrument. The glass has but one motion, but 
the frame has another at right angles to it. 

As can be seen from the illustration, the entire instrument 
can be attached to an object by means of a wood screw, and 
clamped in any position by other screws. In the centre of 
the mirror the silvering is erased, making a small hole through 
which the light of the sun can pass ; also in the centre of 
the frame carrying the mirror there is an opening fitted with 
a convex lens, and behind the lens is a white reflecting sur- 
face — usually chalk. To use the heliotrope, turn the glass so 
that the bright point caused by the sun shining through the 



48 



GEODETIC OPERATIONS. 



hole coincides with the opening in the frame. This will give 
in the focus of the lens an image of the sun, which will be re- 
flected back through the hole in the glass. Now, 
if the entire instrument be turned so as to bring 
this image upon the point at which the light is 
to be seen, the rays falling upon the mirror will 
be reflected in the same direction. 

To see the fictitious sun, as the image is called, 
one must look through the hole from behind the 
glass, and as it is always small and quite indis- 
tinct, some practice will be needed to recognize 
it. This can best be acquired by turning the 
image upon the shaded side of a house, then it 
will be seen as a small full moon. The reflect- 
ing surface can be moved in or out by a screw 
from behind, and the only adjustment that is ever needed 
is to have this surface at that distance that gives the best 
image of the sun. After having placed the heliotrope in 
the correct position, it should be clamped, and then the only 
labor is simply to occasionally turn the glass so as to bring 
the bright spot into coincidence with the opening in the frame. 
In the Eastern States, through air by no means the clearest, a 
light from a Steinheil heliotrope has been observed upon at a 
distance of 55 miles. 

They are made by Fauth of Washington. 




Fig. 5. 



BASE-MEASUREMENTS, 49 



CHAPTER III. 
BASE-MEASUREMENTS. 

As the foundation of every extended scheme of trigonomet- 
ric surveys must be a linear unit, it is essential that the length 
of this base should be determined with the utmost degree of 
care. 

But the labor and expense of measuring a base of favorable 
length are so great as to preclude repeated measurements. In 
order, therefore, to secure results at all comparable with the 
precision desired, an apparatus of great delicacy is needed. 
This becomes apparent when we consider that an apparatus of 
convenient length is repeated from one to two thousand times 
in the measurement of a base, and that even a small error in 
the length of the measuring unit will be multiplied so as to 
seriously affect the results. 

And this error in a short line will be increased proportion- 
ally in the computed lengths of the long sides of the appended 
triangles. The figure and magnitude of the earth are deter- 
mined from extended geodetic operations, and the elements so 
determined are conditionally used in the re-reduction of trian- 
gulation data, securing in this way a more probable expression 
for the shape of our planet. 

From this it may be seen that all of our errors are of an ac- 
cumulative character, and seriously affect the results unless 
fortuitously eliminated by a principle of compensation. 

Since geodesy first received attention, the subject of most 
important consideration has been the construction of a base- 
apparatus that would secure good results without sacrificing 



50 GEODETIC OPERATIONS. 

time and expense. The first form consisted of simple wooden 
bars, resting on stakes previously levelled, and placed end to 
end. When the configuration of the ground made it necessary 
to make a vertical offset, it was done by means of a plumb-line. 
Another form similar to this had a groove cut in the under 
side to rest upon a rope drawn taut from two stakes of equal 
elevation. In place of laying the rods on stakes or on a catenary 
curve, it was once found convenient to place them on the ice, as 
when Maupertuis measured the base in Lapland in 1736. This 
line was measured twice, each time by a different party ; the 
difference between the two results was four inches. This was 
close work in a measurement extending over a distance of 8.9 
miles. The rods used in this case were thirty-two feet long, 
made of fir and tipped with metal to prevent wearing by attri- 
tion. The Peru base measured at about the same time gave a 
difference of less than three inches in the two measurements 
in a distance of 7.6 miles. The wooden rods were found to 
be affected by changes in the hydrometric conditions of the 
atmosphere. This change was diminished by painting them. 
Finally wood was abandoned as the material, and glass tubes 
substituted. Of course with glass there was a continual change 
in length due to expansion or contraction by thermal varia- 
tions, that was not perceptible in the case of wood, but know- 
ing the rate of expansion, the absolute length at any tempera- 
ture can be theoretically computed. The temperature of each 
tube during the entire measurement was ascertained by the 
application of a standard thermometer, and the length of the 
whole base was reduced to a temperature of 62 Fahr. The 
difficulty of determining the temperature of the tubes was 
considerable, since the thermometer reading gives the temper- 
ature of the mercury in the thermometer, or, at best, that of 
the external air, which will always differ from the temperature 
of the measuring-bar. In the case of a sudden change of tem- 
perature, the thermometer will respond more quickly than the 



BA SE-MEA SUREMENTS. 5 1 

tubes, and its reading could not be taken as the reading of the 
tubes. This trouble suggested the construction of an appara- 
tus that would serve to indicate change in temperature — as a 
metallic thermometer. On this principle, Borda made four rods 
for the special committee of the French Academy in 1792. 
The rods were made of two strips of metal — one of platinum, 
and the other of copper overlying the former. They were 
fastened together at one end, but free at the other and through- 
out the remaining length. The copper was shorter than the 
platinum by about six inches. It carried a graduated scale, 
moving by the side of a vernier attached to the platinum ; the 
reading of the scale indicated the relative lengths of the two 
strips, and hence the length and temperature of the platinum. 
The strips rested upon a bar of wood — the entire apparatus 
being six French feet in length. Contact was made by a slide, 
the end of which was just six feet from the opposite end of the 
platinum strip when the zero-mark on the slide coincided 
with one on the end of the strip to which it was attached. 
The rods rested upon iron tripods with adjusting-screws for 
levelling, and the inclination was ascertained from a sector 
carrying a level. It is interesting to note that the length of 
the metre was first determined from the length of the quad- 
rant computed from the base measured with this apparatus. 
Borda's compensating apparatus in some form has been used 
ever since it first came into notice. The principal varieties 
are : Colby, Bache-Wurdeman, Repsold, Struve, Bessel, Hos- 
sard, Borden, Porro, Reichenbach, Baumann, Schumacher, 
Bruhns, Steinheil. 

In these varieties — named after their inventors or improvers 
— the essential features sought for are : 

1. The terminal points used as measuring-extremities must, 
during the operation, remain at an unvarying distance apart, 
or the variations therefrom must admit of easy and accurate 
determination. 



52 GEODETIC OPERATIONS. 

2. The distance between these extremities must be com- 
pared with a standard unit to the utmost degree of accuracy, 
and the absolute length determined. 

3. In its construction provision must be made to secure 
readiness in transportation, ease and rapidity in handling, 
stability of supports and accuracy in ascertaining exact con- 
tact and inclination. 

The above conditions were secured in a great degree in the 
Bache-Wurdeman apparatus, as used in the U. S. Coast and 
Geodetic Survey since 1846. The description given by Lieu- 
tenant Hunt in 1854 will be found quite explicit. For the 
benefit of those who cannot consult the report which contains 
this description the following abstract is given : the apparatus 
sent to the field consists of two measuring-tubes exactly alike, 
each being packed for transportation in a wooden box ; six 
trestles for supporting the tubes — three being fore trestles and 
three, rear trestles — each of which is packed in a three-sided 
wooden box ; eight or more iron foot-plates on which to place 
the trestles, and a wooden frame is afterwards made to serve 
as a guide in laying down the foot-plates ; a theodolite for 
making the alignment, and for occasionally referring the end 
of the tube to a stake driven in the ground for the purpose ; a 
standard six-metre bar of iron in its wooden case, and a Sax- 
ton pyrometer for effecting a comparison. 

The measuring-bar consists of two parts — a bar of iron and a 
bar of brass, each less than six metres in length. 

These are supported parallel to each other ; at one end are 
so firmly connected together by means of an end-block, in 
which each bar is mortised and strongly screwed, as to preserve 
at that point an unalterable relation. The brass bar, which has 
the largest cross-section, is sustained on rollers mounted in 
suspended stirrups ; the iron bar rests on small rollers which 
are fastened to the iron bar, and run on the brass one. Sup- 
porting-screws through the sides of the stirrups are adjusted to 



BASE-MEASUREMENTS. 53 

sustain the bars in place, and also serve to rectify them. Thus, 
while the two bars are relatively fixed at one end, they are 
elsewhere free to move ; and hence the entire expansion and 
contraction are manifested at one end. The difference in the 
length of the two bars is read on a scale attached to the iron 
bar by means of a vernier fastened to the brass bar. The 
scale is divided into half millimetres, of which the vernier indi- 
cates the fiftieth part, so that by means of a long-focus micro- 
scope the difference may be read to the hundredth part of a 
millimetre without opening the case. Since the compensation 
(described further on) can be made correct within its thirtieth 
part, it is evident that the true length of the compound bars 
may be known at any time from the scale-reading, with an un- 
certainty no greater than the thousandth part of a millimetre 
or a microm. 

The medium of connection between the free ends of the two 
bars is the lever of compensation, which is joined to the lower 
or brass bar by a hinge-pin, around which it turns during 
changes of temperature. A steel plane on the end of the iron 
bar abuts against an agate knife-edge on the inner side of the 
lever of compensation. This lever terminates in a knife-edge, 
turned outward at such a distance from the centre-pin and the 
other knife-edge bearing, that the end edge will remain un- 
moved by equal changes of temperature in the two bars. The 
end edge presses against a steel face in a loop made in the 
sliding-rod. This rod slides in a frame fastened to the top of 
the iron bar, and passes through a spiral spring, which acts 
with a constant force to press the loop against the knife-edge. 
The outer end of the sliding-rod bears the limiting agate plane. 
Thus the end agate is not affected in position by the expan- 
sions of the brass and iron, acting as they do at proportional 
distances along the lever of compensation, measured from its 
sliding-end bearing. The rates of expansion for iron and brass 
may safely be taken as uniform between the extreme expan- 



54 GEODETIC OPERATIONS. 

sions and contractions to which they are subject in practice, and 
the compensating adjustment once made is permanent. 

The stirrups sustaining the rollers on which the brass bar 
runs are made fast to the main horizontal sheet of the iron 
supporting and stiffening work. This consists of a horizontal 
and a vertical plate of boiler-iron, joined along the middle line 
of the horizontal sheet by two angle-irons, all being perma- 
nently riveted. Circular openings are cut out from both plates 
to lighten them as much as practicable. A continuous iron 
tie-plate, turned up in a trough-form, connects the bottoms of 
all the stirrups. At the ends, stiffening braces connect the 
two plates. 

We now pass from the compensating to the sector end of the 
tube, at which extremity are arranged the parts giving the 
readings, and for adjusting the contacts between successive 
tubes in measuring, thus making it the station of the principal 
observer. The sector-end terminates in a sliding-rod, which 
slides through two upright bars, and at its outer end bears a 
blunt agate knife-edge, horizontally arranged, which in measur- 
ing is brought to abut with a uniform pressure against the 
limiting agate plane of the compensating end of the previous 
tube. At its inner end, this sliding-rod rests against a cylindri- 
cal surface on the upright lever of contact, so mounted as at its 
bottom to turn around a hinge-pin. At top, this lever rests 
against a tongue, or drop-lever, descending from the middle of 
the level of contact, which is mounted on trunnions.* The 
sliding-rod, when forced against the side of the lever of contact, 
presses its top against the tongue of the level, and thus turns 
the level by overcoming a preponderance of weight given to its 
farther end, to insure the contact being always at a constant 

* The device of the level of contact is supposed to be due to the elder Repsold, 
who applied it first to the comparing-apparatus used by Bessel, in constructing 
the Prussian standards of length. A duplicate of that comparator was procured 
for the Coast Survey, by F. R. Hassler, Superintendent, in 1842. 



BA SE-MEA SUREMENTS. 



55 



pressure between the agates, the same force being always re- 
quired to bring the bubble to the 
centre. The arrangement at the 
two ends is shown in Fig. 6. 

The sector is a solid metal 
plate, mounted with its centre of 
motion in the line of the sliding- 
rod, and having its arc graduated 
from a central zero to the limits 
of ascending and descending 
slopes on which the apparatus is 
to be used. A fixed vernier in 
contact with the arc gives the 
slope-readings. A long level and 
bubble-scale are so attached and 
adjusted to the face of the sector- 
plate that the zeros of the level 
and of the limb correspond to 
the horizontal position of the 
whole tube. If, then, on slopes, 
the bubble be brought to the 
middle by raising or lowering the 
arc-end of the sector (a move- 
ment made by a tangent-screw, 
whose milled head projects above 
the tin case of the tube), the 
vernier will give the slope at 
which the tube is inclined, and 
the sloping measure is readily 
reduced to the horizontal by 
means of a table prepared for 
the purpose. The level of con- 
tact and the lever of contact, 
with their appendages, are all mounted on the sector and par- 
take of its motions. A knife-edge end of the sliding-rod presses 




56 GEODETIC OPERATIONS. 

on the cylindrical face of the contact-lever, this cylinder being 
concentric with the sector, and the sector can therefore be 
turned without deranging the contact. In fact, the contacts 
are made with the sector-level horizontal, thus insuring the ac- 
curacy of the contact-pressure. The contact-lever is supported 
at bottom by two braces dropping down from the sector-plate, 
and a spring, acting on a pin in the lever, steadies it against an 
adjusting screw-end. A bracket from the sector-plate receives 
the trunnions of the contact-level. A small screw projects from 
the end of the tube to clamp or set the lever and level of con- 
tact against a pin in the sector for security in transportation. 

What is called the fine motion, required for adjusting the 
contacts between the successive tubes, is produced by means 
of a compensating rod or tube, one end of which is attached to 
the truss-frame by a bracket over the rear trestle, and the other 
receives a screw terminating in a projecting milled head. This 
screw turns freely in a collar, bearing, by a projecting arm, 
against the cross-bar which joins the main brass and iron bars, 
and its nut is in the end of the compensation-rod. By turning 
the screw in one direction, the bars are pushed forward, and 
the opposite turning permits a spiral spring, arranged for the 
purpose, to push back the system of bars, which slides through 
its supports. Thus the contact is made by turning the screw 
until the contact-level is horizontal. The compensating-rod 
is composed of several concentric tubes, alternately of brass 
and iron, arranged one within the other, and fastened at oppo- 
site ends alternately. Thus, when a contact has been made by 
the fine-motion screw, changes of temperature will not produce 
derangement, as would be the case if this rod were not com- 
pensating. The arrangement permits the observer conveniently 
to work the fine-motion screw, and to observe its action on the 
contact-level. 

The apparatus thus described is enclosed in a double tin tubu- 
lar case, diaphragms being adapted for supporting and strength- 
ening the whole. The air-chamber between the two cases, one 



BASE-MEASUREMENTS. 57 

and a half inches apart, is a great check on heat-variations. 
Three side-openings, with tin and glass doors in each tube, 
permit observations of the parts and of inserted thermometers. 
The ends are closed, only the sliding-rod ends projecting at 
each extremity, exposing the agates. Brass guard-tubes pro- 
tect these, and for transportation tin conical caps are screwed on 
the tube-ends. The fine-motion screw, the sector-tangent screw, 
and the contact-lever-clamp screw project beyond the case. 
The tube is painted white, which, with the air-chamber and 
thorough compensation, effectually obviates all need of a screen 
from the sunshine, which has usually been deemed requisite. 

The tube rests on a fore trestle and rear trestle, which are 
alike, except in the heads. Each trestle has three legs, com- 
posed of one iron cylinder moving in another by means of a 
rack, pinion, and crank, so as to raise or sink the head-plate. 
The levelling and finer adjustment are by means of a foot- 
screw in each leg, by working which a circular level on the 
connecting-frame is adjusted. A large axis-screw, resting on 
the connecting-frame, and rising into a tubular nut, is turned 
by bevelled pinions worked by a crank, and thus raises or lowers 
this tubular nut and the cap-piece which it supports at top. 
The axis-screw, the leg-racks, and the foot-screws give three 
vertical movements in the trestle, by which its capacity for 
slope-measurements is much amplified. 

In the cap of the rear trestle, a lateral and a longitudinal 
motion are provided for, by means of two tablets arranged to 
slide, the upper one longitudinally on the lower one, and the 
lower laterally on the head-plate of the axis-screw tube. Long 
adjusting screw-handles extend to the observer's stand from 
these two plates and from the axis-screw, enabling him to raise 
or lower, to slide forward or back, to the right or the left, the 
rear end of the tube. The fore trestle is similar, except that 
its head is only arranged for a lateral movement, and a second 
observer makes its adjustments by a simple crank. 

Four men can carry a tube, by levers passed through staples 



58 



GEODETIC OPERATIONS. 



in blocks strapped under the tubes. The principal observer 
and an assistant make the contacts and rectifications, the first 
assistant directs the forward tube, and another preserves the 
alignment with a theodolite. A careful recorder notes down 
the observations, and an intelligent aid places the trestles and 
foot-plates. 

This scale referred to, known as Borda's scale, was introduced 
in Bessel's system, the only difference being that he used iron 
and zinc in the place of copper and platinum, and measured 
the interval with a glass wedge. In this the iron is the longer, 
and supports on its upper surface the zinc. 

The zinc terminates at its free end in a horizontal knife-edge, 
and the iron bar very near this has attached to itself a piece of 
iron with a vertical knife-edge on each side in the direction of 
the length of the bar. The distance between the end of the 
zinc and this fixed point, changing with the varying tempera- 
ture, is measured by means of a glass wedge, whose thickness 
varies from 0.07 of an inch to 0.17 of an inch, with 120 divis- 




FlG. 7. 

ions engraved on its face, the distance between its lines being 
0.03 of an inch. The other vertical knife-edge, projecting 
slightly beyond the end of the bar, is brought, in measuring, 
very near the horizontal knife-edge in which the opposite end 
of the bar terminates, and the intervening distance measured 
with the same glass wedge. If the wedge in this case be care- 
fully read and its thickness at each division accurately known, 
this method eliminates some of the uncertainties in the method 
of contact. A pair of Bessel bars, slightly modified, is now in 
use in the Prussian Landes-triangulation. 



BA SE-MEA S UREMENTS. 5 9 

The annexed cut shows the arrangement of the knife-edges 
in the two ends of the Bessel bars. 

The apparatus devised by Colby consists of a bar of brass 
and one of iron, fastened at their centres, but free to move the 
rest of their lengths. Each end of one of the bars is a fulcrum 
of a transverse lever attached to the same end of the other bar, 
the lever arms being proportional to the rates of expansion of 
the bars. In this way the microscopic dots on the free ends of 
the levers are theoretically at the same distance apart for all 
temperatures. As the terminal points were the dots on the 
lever arms, contact could not be made in measuring, so the in- 
terval between two bars was determined by a pair of fixed micro- 
scopes at a known distance apart. 

In all forms of compensating-bars, the components having 
different rates of heating and cooling, their cross-sections should 
be inversely proportional to their specific heats, and should be 
so varnished as to secure equal radiation and absorption of 
heat. Struve's apparatus consists of four bars of wrought iron 
wrapped in many folds of cloth and raw cotton. 

Contact is made by one end of a bar abutting against the 
the lower arm of a lever attached to the other, while the upper 
arm passes over a graduated arc on which a zero-point indi- 
cates the position of the lever for normal lengths. The tem- 
perature is ascertained from two thermometers whose bulbs lie 
within the bar. 

From these descriptions it can be seen that the Bache appa- 
ratus was a combination of principles separately used before. 
It had Borda's scale, Colby's compensation-arm, and Struve's 
contact-lever; with this difference: the lever, instead of sweep- 
ing over a graduated arc, acted upon a pivoted level. The 
form used by Porro in Algiers consisted of a single pair of bars 
attached at their common centre and free to expand in both 
directions. Each end of one of the bars carried a zero-point, 
while the corresponding end of the other had a graduated scale 
like Borda's. In measuring, a micrometer microscope is placed 



60 GEODETIC OPERATIONS. 

on a strong tripod, with an adjustable head immediately over 
the initial point. The apparatus is then placed in position on 
another pair of trestles, completely free from the microscope- 
stands, and moved by slow-motion screws until it is in line 
and the zero-point in the axis of the microscope. The scale is 
then read by means of the micrometer ; at the same time another 
similar microscope is being adjusted over the forward end and 
read. The bar is then carried forward, placed in position so 
that its rear end is under the second microscope, and the for- 
ward end ready for a third microscope previously aligned. 

And so the work progresses until a stop is to be made ; then 
the bar is removed and a point established under the forward 
end of the bar. Every precaution is taken to estimate flexure 
and to avoid uncertainties of collimation and unstable micro- 
scopes. In Ibafiez's apparatus the component bars are copper 
and platinum, mounted upon a double T-iron truss. Flexure 
is determined by resting a long level on the bars at several 
points at equal distances apart. It differs from the preceding 
in having the bars exposed. 

The Baumann apparatus, recently constructed for the Prus- 
sian Geodetic Institute, has platinum and iridium bars resting 
on an iron truss, with its entire length open to the free circula- 
tion of the air. Inclination is determined by a level of preci- 
sion and flexure by a movable level. 

There are six microscope stands, the same number of trestles 
for the bars, and thirty sets of heavy iron foot-plates. 

The latter are put in position, and remain half a day before 
being used. For each microscope-stand there are two tele- 
scopes — one for aligning and one for reading the scales. Six 
skilled observers and about thirty laborers are needed in meas- 
uring. Only one base has been measured with this apparatus 
up to the present time — that of Berlin in 1884 — but the results 
are not yet known. 

The Repsold differs from the Baumann apparatus only in a 
few points, the chief being: the component bars are steel and 



BA SE-MEA S UREMENTS. 



6l 



zinc, and the two are suspended in a steel tube which is 
wrapped in thick felt. The small probable errors deduced by 
Ibanez and the officers of the Lake Survey in the results ob- 
tained with the metallic-thermometer principle appear to com- 
mand its continuance in the construction of base-apparatuses. 
But in the Yolo base authenticated temperature changes were 
not always accompanied by corresponding indications of the 
Borda scale. In short, the behavior of the zinc component was 
so unsatisfactory that a new apparatus for the Coast and Geo- 
detic Survey is under consideration, in which the scale-readings 
will be omitted, and either a partly compensated pair of bars 
or a single carefully protected bar of steel adopted instead, 
with daily comparisons with a field standard. For additional 
information on the various forms of base-apparatuses the au- 
thorities cited at the end of this chapter may be consulted. 

It is interesting to note the results of various measurements 
under different auspices with the same or different forms of 
apparatuses. The following list gives the most important: 



Name of base. 

Dauphin Island 
Bodies Island. . 
Edisto Island .. 
Key Biscayne. . 

Cape Sable 

Epping Plains.. 
Peach Ridge. .. 
Fire Island .... 

Kent Island 

Beverloo 

Ostend 

Cape Comorin.. 

Keweenaw 

Minnesota 

Chicago 

Sandusky 

Wingate 

Yolo 

Aarberg 

Weinfelden 

Toederen 

Ilidze 

Speyer 

Fog?ia 

Naples 

Axevalla 



Measured by. 



Nerenberg. 



Eng. Trig. S 
U.S. LakeS. 



Apparatus. 



U.S. C. and G. S Bache-W 



U. S. Geol. S 

U. S. C and G. S 

Ibanez and Hirsch 

Hirsch 

Haffner and Overgaard . . 

Kalmer and Lehrl 

Schwerd 

Italian Government 



Stecksen. 



Hassle r. 
Bessel . . 



Colby . . 
Bache-W 

Repsold . 



Slide-contact 

Davidson 

Ibanez 



Swedish Acad'y. 

Austrian 

Schwerd 

Bessel 



Wrede 



Length. 



6.66 

6-75 

6.66 

3-6 

4 

5-4 

5-8 

8-75 

5-5 

2300 

2480 

8912 
5 
3 
4 
3 
4 

17486 

2400 

2440 

33i8 

4061 

859 
2016 

340 
J 357 



les. 



feet, 
miles. 



51 : 

07 

29 

55 
34 

44 

569 

224 

033 



Prob. error 



41000 

425500 

418600 

454400 

409600 

551600 

5618800 

483980 

22800 

16949 

22222 

667000 

83310 

530000 



1 148600 

54366 

700000 

6000000 

3500000 

2090000 

3700000 

7 15000 

13^3065 

963784 

x 577945 



62 



CE ODE TIC OPERATIONS. 



Perhaps a better idea can be obtained of the accuracy of 
base-measurements when we give a comparison of the measured 
length of a line with its length as computed from another base. 
A few such comparisons are here given : 

Epping measured 871 5.942 metres. 

Computed from Massachusetts base 871 5.865 " 

" " Fire Island base 8715.900 " 

Massachusetts base measured 17326.376 " 

Computed from Epping base 17326.528 " 

" Fire Island 17326.445 " 

Combining the errors of preliminary measurements with the 
computed error in the triangulation, the appended results are 
obtained: 



Probable error in junction-line. 


Due to base. 


To triangulat'n 


Both. 


From Epping base 


0.17 metre. 
0.20 " 
o.39 " 


0.76 metre. 
0.32 
0.66 " 


0.78 metre. 

o.37 " 
0.77 " 


'* Massachusetts base 







Considering the distance apart of these bases, it is safe to say 
that if the errors are constant the maximum error in the length 
of any line of the triangulation is not more than 0.22 of an 
inch to the statute mile. The above are the results of measure- 
ments by the Bache-Wurdeman apparatus, angles measured 
with a thirty-inch repeating-theodolite, and the triangulation 
computed by Mr. Schott. Simply with the purpose of com- 
paring the results obtained by different apparatus, I make an 
extract from the report of the U. S. Lake Survey : 

Chicago base measured log. in feet 4.3917929 

" " computed from Fond du Lac " " 4.3918010 

Difference = 0.14 metre. 
Distance from Chicago to Fond du Lac, 150 miles. 



BASE-MEASUREMENTS. 



63 



Olney base measured log. in feet 4.334923 1 

" " computed from Chicago " " 4.3349231 

Difference = 0.06 metre. 
Distance from Chicago to Olney, 200 miles. 



The Madridejos base, measured by General Ibanez with his 
improved Porro apparatus, was divided into five segments; the 
central one was about 1.75 miles long. This one was meas- 
ured twice, and used as a base in computing the length of each 
of the other segments. The relation between the measured 
and computed values may be seen in the following table : 



Segment. 


Measured (metres). 


Computed (metres). 


Difference (metres). 


I 

2 

3 

4 

5 

Total 


3077.459 
2216.397 
2766.604 
2723.425 
3879.000 


3077.462 
2216.399 
2766.604 
2723.422 
3879.002 


— 0.003 

— 0.002 

+ 0.003 

— 0.002 


14662. S85 


14662.889 


— 0.004 





The Wingate base, measured with a slide-contact apparatus, 
was divided into three segments; the middle one was measured 
twice to see if a discrepancy sufficiently great to warrant a re- 
measurement existed. The two results were in sufficient ac- 
cord to admit of the acceptance of the entire measurement as 
correct. However, each segment was used as a base for the 
computation of the other segments. 

The length of the line was : 



With measured first and computed 2d and 3d. . . 6724.5309 m. 

" second " 1st and 3d. .. 6723.7132 " 

" third " 1st and 2d.. . 6723.8248 " 

Measured value of the whole line 6724.0844 " 



64 GEODETIC OPERATIONS. 

Giving these values equal weight, the length may be written 
6724.0383 ± 0.12 metres. 

Colonel Everest, with the Colby apparatus, measured in India 
three bases, and joined them in the scheme of triangulation, 
measuring the angles with a thirty-six-inch theodolite. 

Dehra Dun. Damargida. 

Measured length in feet 39183.87 41578.54 

Computed- 4 ' " 39183.27 41578.18 

Only instructions of the most general kind can be given for 
the mechanical part of measuring. The details vary with each 
form of apparatus. The location of the base is a matter of 
prime importance, and must be considered in connection with 
the purpose for which the base is needed. If for verification, 
it should be suitably situated for connection with the chain of 
triangles it is intended to check. 

If it is intended to serve as an initial base, a favorable con- 
dition for immediate expansion should be sought. As the 
base will usually be from three to seven miles long, the points 
suitable for the first triangle-stations should be somewhat 
farther than that apart, permitting a gradual increase in the 
lengths of the sides. The best initial figure is undoubtedly a 
quadrilateral of which the base is a diagonal, giving an expan- 
sion from either side, or from the other diagonal. 

If this be impracticable, the base must be a side of a com- 
plete figure. Of course the termini must be intervisible, and 
at the same time visible from every point of the line. If the 
ground is irregular, having slopes exceeding three degrees in 
inclination, it must be graded to within that limit, with a 
width of about twelve feet. The method of alignment varies 
with the views of the person in charge. 

A good plan is to select a point approximately at the mid- 
dle of the line. Place a theodolite there, and direct the tele- 



BASE-MEASUREMENTS, 6$ 

scope to the temporary signal at one end and read the angle 
to the other end ; if it differs from 180 , move the instrument 
in the proper direction until the angle is just 180 . Assistants 
are then sent towards each end, and, from signals from the per- 
son at the instrument, secure points in line: these should be 
placed about a quarter of a mile apart. Considerable experi- 
ence has shown that the best form of aligning signal is a piece 
of timber of suitable size, 2x4 inches or 4 inches square, driven 
in the ground and sawed off a few inches above the surface. 
In the top of this, bore a hole at the central point for the 
insertion of an iron pin, twice as long as the hole is deep. 
Take a corresponding piece of timber six or eight feet long and 
make a similar hole in its end. It can then be adjusted to the 
stake in the ground, and made stable by two braces, after be- 
ing made perpendicular by means of a plumb line or a small 
theodolite. The advantage of this form of signal is that it can 
be removed when the measuring reaches this point, and be re- 
placed for a future measurement without going to the trouble 
of making a second alignment. A plan of aligning differing 
from this is to have the instrument carefully adjusted and 
placed three or four hundred yards from the end. Direct the 
telescope to the temporary signal at that point, turn it in its 
Y's, or 180 in azimuth, and fix a point directly in line. Then 
place the instrument over the point so selected and locate 
another point in advance, and so on till the opposite end is 
reached. This will only be possible when one terminus has 
been decided upon and the general direction of the line. Each 
terminus of the base is marked by a heavy pier of masonry of 
secure foundation with upper surface eighteen inches or two 
feet below the surface of the ground. In the centre of the large 
stone forming a part of the top of the pier a hole is drilled ; in 
this, with its upper face even with the top of the stone, is 
placed, and secured by having poured around it molten lead, 
a copper bolt or a piece of platinum wire. 
5 



66 GEODETIC OPERATIONS. 

On the upper end of this bolt or wire a needle-hole may be 
drilled, or a pair of microscopic lines drawn, whose intersection 
marks the end of the base. Immediately above this should 
be placed a surface-mark to which the position of the theodo- 
lite can be referred in the triangulation ; also a set of witnesses 
consisting of four stones projecting above ground, so placed 
that the diagonals intersect above the under-ground mark. 

When both ends are marked in this way before measuring, 
the distance from the end of the last bar to the terminal 
mark, already fixed, is measured on a steel scale horizontally 
placed. 

The only advantage possessed by this method is, that both 
monuments have an opportunity to settle before the distance 
between them is determined. It is believed, however, that 
greater inaccuracies will result from the uncertainty in this 
scale and its use than from the irregular settling of the pier 
placed after the measurement is finished. Before beginning the 
accurate measurement it is advisable to make a preliminary 
measurement with a steel tape or wire, marking every hun- 
dred lengths of the apparatus to be used. This will serve as a 
check upon the record as the final work advances ; and if the 
line is to be divided into segments it will show where the in- 
termediate monuments are to be erected. When these inter- 
mediate stations are occupied the angle between the ends and 
the other points should be measured with great care, so that, 
if the line be found to be a broken one, the exact distance be- 
tween the termini in a straight line can be computed. If the 
required distance cannot be obtained without crossing a ra- 
vine or marsh, the feasible parts can be measured, and the other 
portion computed by triangulation. 

The form of record will of course vary with the kind of ap- 
paratus used, but too much care cannot be taken in keeping 
the record. The principal data needed in the reduction may 
be stated as follow : 



BASE-MEASUREMENTS. 6? 

1. The time— showing the time at which each bar was placed 
in position in order to form some idea of the average speed at- 
tained in the work. 

2. The whole number of the bar. When a preliminary 
measurement has been made as suggested, the hundredth bar 
should end near the stake previously driven; if not, a remeas- 
urement must be made from the last authentic point. This 
should be at the end of the even-hundred bar, and perhaps 
more frequently, especially if the day should be windy, en- 
dangering the stability of the bars, or if the ground should be 
boggy or springy. The simple method for placing this point 
is to set a transit or theodolite at right angles to the line and 
at a distance of twenty-five or thirty feet from it. After level- 
ling, fix the cross-wires of the instrument upon the end of the 
bar; then, pointing the telescope to the ground, direct the 
driving of a stake in a line with this and with the aligning 
telescope. The height of the telescope should be half the 
height of the bar, so that the focus need not be changed. 

Then in the top of this stake a copper tack is driven, and on 
its upper face are drawn two lines coinciding with the vertical 
threads of the two instruments. If they are in good adjust- 
ment the intersection of these lines will mark the end of the 
bar. A record must always be made when a stub is thus 
placed. It is also advisable to place a stub under the instru- 
ment used for this horizontal cut-off, so that if it should be 
necessary to begin work at this point the instrument would 
occupy the same position that it occupied before, eliminating 
by this means the error that would arise from not having the 
transit at right angles to the line. 

Probably a more accurate method is to have a metal frame 
one inch wide and two inches long with screw holes admitting 
of attachment to a stake. This frame has sliding inside of it 
another that can be moved by a milled-head screw, with a set 
screw to hold it in place. On the upper surface of this frame 



68 GEODETIC OPERATIONS. 

is a small dot or hole. When the approximate position of the 
end is determined by a plummet, a stake is driven in the ground 
until only an inch or so remains above the surface: to this is 
attached the outer frame ; then, with the theodolite previously 
set upon the end of the measuring-bar, direct the movement 
of the inner frame until the hole or dot is bisected by the 
cross-wires, when the frame is clamped in place and verified. 
When microscopes are used, the dot can be brought under the 
micrometer-wire that marked the position of the zero-point on 
the bar. 

3. The designation of the bar as A, B, or 1, 2, etc., so that 
it may be known how many times each bar was used. Since 
the two are never of the same length, the distance obtained 
by each bar must be separately computed and the two values 
added to get the entire length of the line. 

4. Inclination. When going up-hill the inclination is recorded 
plus, and minus when going down. However, as the correc- 
tion for inclination is always subtracted, the sign is of small 
consequence. 

5. Columns for the sector-error and the corrected values for 
the inclination. Before beginning work each day the rods 
should be placed on their tripods and be made perfectly hori- 
zontal by raising one of them. To determine this, set up a 
carefully adjusted theodolite at such a distance that both ends 
of the bar can be seen. Set the thread on one end of the bar, 
revolve the instrument in azimuth, and see if the thread be 
on the other end: when such is the case, bring the bubble of 
the sector in the middle of the tube and see what the scale- 
reading is ; if zero, then there is no error. This test should be 
applied at the beginning and close of each day's work, and the 
average error added to or subtracted from the reading of in- 
clination for that day. With secondary apparatus this is un- 
necessary, as the positive and negative readings will be about 
equal, so that the number of readings that are recorded too 



BASE-MEASUREMENTS. 69 

great will be corrected by those that are too small by the same 
quantity. 

6. Temperature. The thermometers should be read about 
every ten bars, and in the Borda rods the scales more fre- 
quently. When the temperature gets above 90 Fahr., it is 
advisable to stop work, especially if the bars are not compen- 
sated, as the adopted coefficients of expansion at that tem- 
perature are unreliable. 

The Repsold apparatus, as used on the Lake Survey, and 
the Davidson, with which the Yolo base was measured, were 
protected during measuring by a canopy made of sail-cloth 
mounted on wheels, so as to move along as the work advanced. 

In all kinds of apparatus it is advisable to measure when the 
bars indicate a rising temperature, and also during the time 
required for them to fall through the same amount. 

7. A column for corrections for inclination, computed from 
a formula to be given. 

8. A column for remarks, explaining delays, stoppages, the 
placing of stubs, etc. ' 

GENERAL PRECAUTIONS TO BE TAKEN WHILE MEASURING. 

The rear end of the bar must be directly over the marking 
on the initial monument. 

The inclination must never be so great as to endanger a 
slipping of the bars forward or backward. 

The trestles should be so firmly set that there can be no un- 
equal settling after the bar has been placed on them. 

A bar should not be allowed to remain more than a minute 
in the trestles, lest its weight should change their position. 

When a stoppage is made to allow the aligning-instrument 
to advance, a transit should be set up, as already described, 
and its cross-wires firmly clamped on the end of the bar; then, 
before resuming work, the position of the bar can be restored 
if from any cause it has changed. 

When the end has been transferred to a 'temporary mark, as 



JO GEODETIC OPERATIONS. 

when a stop is made for night or dinner, in resuming work it 
is best to place the bar that the work closed with in the same 
position it had before stopping ; then the new day's work goes 
on as though there had been no break. If this plan is not 
adopted, either in the transference to the ground or from it, 
the end sighted will be more than the standard length from 
the other end, being held out by the spiral spring that keeps 
the agate beyond its proper distance, rendering it necessary to 
record an index-error for every transference ; whereas in the 
plan suggested there can be no danger of omitting to record 
this index-error, nor of recording an erroneous value. 

This precaution refers to that species of apparatus which 
consists of a pair of bars, one abutting against the other, and 
not where only one bar is used, as in the Repsold, Baumann, 
and others. 

The alignment must be made with precision, for all errors of 
this kind are of the same character and do not cancel one 
another. 

Before beginning actual work the party should measure a 
short distance several times, by way of practice, until the dis- 
agreement between two measures is made very small. 

COMPUTATION OF RESULTS. 

In order to know the horizontal distance between the two 
ends of the base it is necessary to know the number of times 
the measuring-unit was used, and its exact length each time 
that it was employed. To this must be added index-errors, 
and the amount by which the last bar fell short of the ter- 
minus. Also, there are to be subtracted the quantities that 
were needed to reduce each length to its horizontal projection, 
and those negative errors that could not be obviated. 

A carefully kept record will show how often the bars were 
used ; but to ascertain their length is a more difficult problem, 
depending upon : (a), a knowledge of the exact length of the 
adopted standard ; (d), a known relation between the measuring- 



BASE-MEASUREMENTS. *J I 

bar and the standard at a certain temperature ; (c), a knowledge 
of the temperature of the bars each time used, and the coeffi- 
cients of expansion. 

The Committee Metre is the standard of linear measures 
now in use, and with a certified copy of this, all our units are 
compared. This comparison can be described only in outline. 
We have two firmly built pillars at a convenient distance apart 
for the bars that are to be compared. On one is an abutting- 
surface, and on the other is a comparator. In general, this 
comparator consists of a pin held out by a spiral spring but 
capable of being withdrawn by a micrometer-screw. This pin 
works a lever on whose longer arm is a point that is to be 
brought into coincidence with a fixed zero-mark. Between 
these two pillars is a carriage rigidly constructed but com- 
pletely isolated from them. On this carriage are placed the 
standard and the bar that is to be compared. The former is 
placed between the abutting-surface and the micrometer-pin, 
the screw is turned until the zero-marks coincide, and the turns 
and division recorded. 

The carriage is then moved along until the bar is brought 
into place and the micrometer is again read. The difference 
in the readings will correspond to the difference in lengths in 
terms of micrometer turns and divisions — the value of a turn 
and a division being found by measuring with the screw the 
length of a standard centimetre. In very accurate comparisons 
the bars are immersed in glycerine, which can be readily kept at 
the same temperature for a long time. 

The temperature is ascertained from three thermometers — 
one at each end, and one at the middle. Also, to eliminate 
accidental errors, a number of readings are made with the bars 
reversed, turned over, taken in different order, and at different 
temperatures. The average difference will be the difference 
in the lengths of the standard and the bar at the average tem- 
perature, supposing that the coefficients of expansion remain 
constant. Then knowing the temperature at which the stand- 



72 



GEODETIC OPERATIONS. 



ard is correct and its coefficient of expansion, its true length 
can readily be computed for this average temperature. To 
this, add the average difference just referred to and we have the 
exact length of our bar at this mean temperature. To illus- 
trate : let Jkfbe the standard, A the bar under comparison, ja 
the difference in microns, which is obtained by multiplying the 
turns and divisions of the micrometer by the previously ascer- 
tained value of one turn. 



Temp. 



M. 



°F. 

57-58 
52.60 
55.29 
55.16 



= /o 



It. 

7-5 
t- 7-5 

9.8 
+ 8.27 



Therefore, A = M-\-8.2?/j. at 55°.i6. Suppose e be the co- 
efficient of expansion for M, and Tthe temperature at which 
Mis correct, then we have A = J/+455°.i6— T) -f 8.27//. 

To determine e we must have the pillars of the comparator 
at a fixed distance apart, and then measure this distance with 
a bar at different temperatures. In order to insure the bar 
being at the same temperature, it is best to place it in glycer- 
ine previously heated, and leave it there for half an hour. Let 
D be the difference between the constant distance and the dis- 
tance as observed at various temperatures, t Q the average, and 
/ the observed temperatures. 



/. 


D. 


t- t . D — A>- 


F. 
99.08 
83.68 
72.08 
57.58 
42.39 
70.95 = h 


441-5 
342-9 
268.2 

175-9 
084.5 
262.6 = Z> 


-\- 28.13,? = 178.9 

12.73^ = 80.3 

i.07<? = 5.6 

- 13.37* = - 86.7 

— 28.56^ = — 178. 1 



BA SE-MEA SUREMENTS. 



73 



Forming the normal equations by multiplying each equation 
by the coefficient of e in that equation, and taking the sum of 
the resulting equations, we get 1948.92^ = 12,306.4/i, or 
e — 6.315 yw. Substituting this value of e, we have for A, 
A = M + 6.3J5yu(55°.i6 — T) + 8.27//. There is a probable 
error in this determination which can be carried through the 
future computations. 

The way in which the temperature-observations are utilized 
depends upon the accuracy desired ; ordinarily the average 
temperature of each bar in a segment is employed. So that if 
we have n lengths of a four-metre bar with the above coeffi- 
cient of expansion, a length equal to A at 5 5 . 16, and the aver- 
age temperature t in that segment, we shall have the distance 
= n[A -f- 4X0.000 oo63i5yu(/— 55°.i6)]. When greater ac- 
curacy is required, the length of each bar can be computed in 
the same manner, and the aggregate length obtained by sum- 
mation. 

When a Borda scale or metallic thermometer is used, it is 
necessary to know how much in thermometric scale a division 
is equal to. The scale is usually divided into millimetres, and 
read by a vernier or microscope to 0.01 mm. 



Temp. 


t-t . 


Scale = S. 


dS. 


°F. 









109.41 


+ 31.79 


8.60 


+ O.92 


94.11 


+ 16.49 


8.17 


+ O.49 


79.21 


+ i-59 


7-74 


-f- O.06 


61.16 


— 16.46 


7.16 


— O.52 


44.22 


- 33-40 


6.72 


— O.96 


77.62 = t 




7.68 = S 





By letting .2: be the quantity representing the differential ex- 
pansion of the component bars, and as it varies with the tem- 
perature, we may take the values of t — t as the coefficients 
of x and solve by least squares. The normal equation will 
give 2671. 54JF = 78.05;/, or x = 0.02922^= 29.22/*. 



7 A GEODETIC OPERATIONS. 

That is, a change of one degree Fahr. is represented by 
0.029 division, or the smallest value that can be estimated 
on the vernier, o.oid = -J°F.; consequently the scale-readings 
can be readily converted into degrees of temperature and the 
reduction for length made as in the preceding case, or the 
change in length may be found directly in terms of scale-read- 
ings. If we have a four-metre bar with the coefficient of ex- 
pansion just found, o.oid = — — jj. = 8.64//. 

Then if 5 be the scale-reading at which M is a standard, 
and 5 any other reading during the measurement or the aver- 
age, A = M-{- 8.64m{S — 5 ), and the entire line 

= u[M+S.64m(S-S )-]. 

Correction for inclination : if R represent the length of a 
bar, h its horizontal projection, and 6 the angle of inclination, 
it is apparent that h = R. cos 6, then d the correction = R — h 
= R - R . cos 6 = R(i - cos 6)=2R. sin 2 %&. As 6 is small 
sin 2 \d = \ sin 2 8 (nearly), so we may write 

^sin 2 6 sin 2 1' sin 2 i' 
d = = RO 2 ; log — = 2.626422. 



Having determined by comparison the average length of the 
bars, a table should be computed for each, giving the values 
for d for each fractional part to which the sector can be read, 
and within the limits observed. Then from this table correc- 
tions for inclination can be taken and inserted in the record- 
book. If there are any index-errors, as stated might occur in 
the transferrence of the end to the ground, they must be added 
to the computed length. 

Probable error. This may be derived — 

I. By measuring the base a number of times, then deducing 



BASE-MEASUREMENTS. ?$ 

the probable error in accordance with the principle of least 
squares. 

2. By dividing the line into segments and computing the 
other segments from each one as a base by triangulation. 

3. By checking one base from another in the chain of trian- 
gulation, and determining the probable error in the second 
from that of the first and of the measurement of the angles in 
the triangulation. 

4. From all known sources of error in measurement. 

The fourth method is the only one that needs expansion at 
this point. The principal sources of error in measurement 
are: 

1. In determining the length of the bar. 

2. Backward pressure. 

3. Error of alignment. 

4. In transferring end to the ground. 

5. In the determination of inclination. 
.6. Personal errors of the observers. 

These are determined as follows : the first is obtained from 
repeated comparisons with the standard, and is made up of 
two parts — uncertainty in the expansion of the bars, and acci- 
dental errors in comparing. Of these the former is found from 
the residuals in the series of determinations of the coefficients 
of expansion. Calling this r/, we have for the entire n bars 
n.r/. Likewise the error from comparison is found in a simi- 
lar manner from the series of comparisons, if we designate this 
r/, the entire error r 2 = n.r a '. 

The error of contact depends upon the force with which the 
agate is held out beyond its proper position. When a bar is 
in its right place, and the next bar brought into contact with 
it, the pressure necessary to bring it to its place forces the rear 
bar backward ; and when the rear bar is taken away the for- 
ward bar, being relieved of this pressure, moves back by the 
same amount. Consequently the total backward movement is 



j6 GEODETIC OPERATIONS. 

double the effect of pressure. This must be determined by 
experiment in various positions of the bar. As every bar ex- 
cept the first and last are doubly affected, these each being 
changed only once by this pressure, the total correction will 
be twice the displacement multiplied by one less than the 
number of bars. Usually this is too small to be considered, 
and applies to those bars only that are used in pairs — one bear- 
ing in contact against the other. 

By (3) is not meant the uncertainty of having the line as a 
whole straight, but in placing the bar exactly in that line. The 
aligning instrument is placed in front at distances varying from 
50 to 900 feet, and the alignment is effected by bringing the 
agate of the bars into coincidence with the vertical thread of 
the telescope ; or when the bars are provided with a vertical 
rod immediately over their centres, this is sighted to. It is 
apparent that the bisection of this may not be perfect ; and, in 
fact, when the light falls unequally upon the object sighted to, 
the illuminated spot is bisected, which may be altogether to 
one side of the centre. 

However, the error of bisection cannot be greater than the 
radius of the agate or aligning-rod, and its effect upon the true 
length of the line will depend upon the distance to the transit. 
The nearer the transit, the less is the likelihood of making an 
erroneous bisection. By placing a scale directly under the 
agate, and having the person at the transit direct the mov- 
ing of the bar until he considers it in line, make a note of the 
scale-reading, and after a number of trials the average variations 
may be taken as the error most likely to be committed at 
that distance. Suppose it was found that the errors were a 
for the maximum, and b for the minimum distances, the an- 
gular variations might be written : a times one second divided 
by the length of the bar, call this/#, and similarly for b, which 
we will call n. The correction for this deviation will be the 
difference between the length of the bar and the vertical pro- 



BA SE-MEA S UREMENTS. J J 

jection for this angular deviation. As already shown, this is 

R . sin 9 #i . R . sin 2 n 
equal to , and . 

Only the first and last few bars of each segment will need 
to have this total lateral correction applied ; for the remaining 
bars it will be sufficient to take the average of m and n, in the 
formulae just given. As the total correction from this cause 
will never amount to a tenth of an inch, it is usually omitted, 
and its probable error is never considered. 

The error from the fourth source is determined from experi- 
ment, as in the preceding case. Suppose it is 0.082 mm.; 
as there is a double transfer, the entire error will be 0.082 
V2 mm. = 0.11 mm., and the total for n bars will be 0.11 
Vn . mm. = r % . 

The fifth source of error is quite apparent. The sector that 
shows the inclination usually reads to single minutes, some- 
times to ten seconds. As it is impracticable to obtain more 
than one reading for each inclination, there is an uncertainty 
as to its correctness. This will vary with the skill of the ob- 
server and the character of the sector used. The probable 
error of a single determination should be ascertained as fol- 
lows : place the bar firmly in its trestles and make several 
readings of the scale when the bubble of the level is in the 
same position. From a number of such scale-readings the 
probable error is deduced in the usual manner. 

To determine the effect of this error on the computed cor- 
rections for horizontal projections, the average observed in- 
clination must be approximated. Suppose this to be 2°, the 
probable error of inclination 30", and the length of the bar R. 
It has already been shown that the correction for inclination d 
= R(i — cos 6). As 6 in this case is taken as 2°, an approxi- 
mate value for the change in d by a mistake of 30" in 6 
can be computed by getting d' when 6 = 6 i 30"; d' = 
R\\ — cos(#,-f- 30")], and the probable error in any one deter- 



78 



GEODETIC OPERATIONS. 



mination will be the difference between d and d' or r/, r K = 
e Vn where n = the number of bars and e = d — d\ 

To recapitulate : those errors that are known to exist and 
the direction of whose effect is unmistakably determined can be 
applied in the reduction of the length of the base, while those 
that are merely probable must be used simply in obtaining the 
probable error of the measurement as a whole. The value for 
the length of the base must be diminished by the amount of 
backward pressure, errors of alignment, and errors of inclina- 
tion ; but the remaining errors having a double sign must be 
regarded as probable ; if individually they be represented by 
r v r v r 27 ' ' ' r n, and the total error by R, we will have 



R = Vr, 2 + r: . . . r n *= V2\f]. 



As the sides of the triangulation are at different elevations 
and the base and check-base not on the same plane, it is neces- 
sary to know their lengths at some common-datum plane. 
This by common consent is the half-tide level of the ocean. 



bg, height above half-tide = h ; 
ae, the half-correction for reduction 



2 » 




ae : ed :: ab : bc\ 

ed .ab , ab 



ae 



be 



2ae 



h. 



2ab 



= h. 



= h. 



cg+bg' 
B 



cg+bg 'cg+bg' 



bg is so small in comparison with eg 
that it may be omitted, and we write ; 



BASE-MEASUREMENTS. 79 

h.B Jk h" 






\R RP 



By-?*— -ts), where R = radius of 



radius of curvature 
curvature at the mean latitude of the base. 

From the corrected value for the length of the base c is to 
be subtracted. If the elevation of the base was found by dif- 
ferent methods, or from different bench-marks, an uncertainty- 
may arise in the value of k, giving a probable error for c. 

REFERENCES. 

U. S. Coast and Geodetic Survey Reports as follows: 1854, 
pp. 103-108; '57, pp. 302-305 ; '62, pp. 248-255 ; '64, pp. 120- 
144; '73, pp. 123-136; '80, pp. 341-344; '81, pp. 357-358; 'S2, 
pp. 139-149; also PP- 107-13.8 ; '83, pp. 273-288. 

Clarke, Geodesy, pp. 146-173. 

Report of U. S. Lake Survey, pp. 48-306. 

Zachariae, Die Geodatische Hauptpunkte, pp. 79-1 jo. 

Jordan, Handbuch der Vermessungskunde, vol. ii. pp. 

73-113- 

Experiences Faites avec l'Appareil a Mesurer les Bases. 

Compte Rendu des Operations de la Commission pour eta- 
lonner les Regies employes a la Mesure des Bases Geodesiques 
Beiges. 

Westphal, Basisapparate und Basismessungen. 

Gradmessung in Ostpreussen, pp. 1-58. 



80 GEODETIC OPERATIONS. 



CHAPTER IV. 

FIELD-WORK OF THE TRIANGULATION. 

SUPPOSING that a base has been carefully measured, or the 
distance between two stations previously occupied accurately 
known, the next thing to be done is to lay out a scheme of tri- 
angles covering the desired territory. Their arrangement into 
figures depends upon : 

1. The special purpose of the work. 

2. The character of the country over which the system is 
to be extended. 

If the object is to measure arcs of a meridian or of a par- 
allel, for the purpose of determining the figure of the earth, 
great care should be exercised in selecting triangles that are 
approximately equilateral ; for if in the computation a very 
long side is to be computed from a short one, an error in the 
latter will be greatly magnified in the former. If the purpose 
is simply to meet the wants of the topographer, the stations 
should be selected with special reference to his needs and 
without regard to the character of the figures thus formed. 
In an open prairie where signals have to be erected without 
any assistance from natural eminences, their arrangement may 
be made in strict accord with theoretical preference. 

The plainest system of the composition of triangles into 
figures is a single string of equilateral triangles which possess 
the advantages of speed and economy of time and labor. Hex- 
agonal figures are preferred by some, but the general prefer- 
ence is for quadrilaterals with both pairs of diagonal points in- 
tervisible. This system covers great area and insures the 
greatest accuracy. 



FIELD-WORK OF THE TRIANGULATION. 8 1 

Equilateral triangles will furnish nine conditions. 

Hexagons, with one side in common, twenty-one conditions. 

Quadrilaterals, twenty-eight conditions, covering the same 
area (approximately). 

Signals. — After deciding upon the positions of the stations, 
the next subject for consideration is the kind of signals to be 
used. In short sights, the best form is either a pole just large 
enough to be seen, or a heliotrope fixed on a stand or a tripod 
carefully adjusted to the centre of the station. As the helio- 
tropers are usually persons with but little experience, range- 
poles should be previously set, enabling them to point their 
instruments with some degree of precision. 

(For a description of the heliotrope, its adjustments, and use, 
see page 45.) 

Owing to the fact that there are so many days during which 
it is impossible to use the heliotrope, and also the additional 
trouble that frequently when the sun is shining the air is so 
disturbed that the object sighted is too unsteady to bisect with 
any certainty, the effort is constantly being made to devise 
some form of night signal to take the place of day signals. 

The great obstacle to the successful solution of this problem 
has been the dimness or expense of the lights that have been 
tried, such as oil-lamps, magnesium, or electric lights. In June, 
1879, Superintendent Patterson of the U, S. Coast and Geo- 
detic Survey directed Assistant Boutelle to make an exhaustive 
series of observations with the various methods of night signals, 
with a view to determine the most effective method to be used 
in triangulation. 

The special points to be considered were : 

1. Simplicity and cheapness. 

2. Adaptability to the intelligence of the men usually em- 
ployed as heliotropers. 

3. Ease of transportation to heights. 

6 



82 GEODETIC OPERATIONS. 

m 

4. Penetration, with least diffraction and most precision of 
definition. 

5. The best hours for observation. 

6. Lateral and vertical refraction, and the extent to which 
the rays are affected by the character of the country over 
which they pass. 

An accurate account of the various experiments made by 
Captain Boutelle are given in Appendix 8 of the C. and G. S. 
Report for 1880. I shall take the liberty of quoting his con- 
clusions ; they are : 

" The experience of the past season enables me to state with 
some precision the cost of the magnesium light, so much supe- 
rior to every other yet tried. 

" The success in two instances of burning the light by a time- 
table established that method as perfectly practicable. 

" It reduces the time of burning it to twenty minutes per 
hour, or to eighty minutes for four hours' observation. With 
a delivery of ribbon of fifteen inches per minute, the cost will 
be two dollars per night for each light used. The average 
number of primary stations observed upon at any one station 
is six, of which three would require the magnesium light, 
making the expense six dollars per night. The nights when 
observation would be practicable and the lights burned maybe 
taken as averaging three in a week, or seven at each station. 

" Apart from the first cost of apparatus, we should therefore 
have as the additional outlay for night observation for a pri- 
mary triangulation : 

" I. Additional pay of six heliotropers $3.00 

"2. " cost of burning three magnesium lights 

every other night 3.00 

''3. " cost of kerosene-oil for three lamps. .. 0.20 
"4. " cost per day for supplies, etc 0.80 

" Whole additional daily cost $7.00 



FIELD-WORK OF THE TRIANGULA TIOiV. 83 

" To offset this additional party expense there will be : 

" 1. The shortening of time required in occupation of each 
station by the addition of four hours of observing each clear 
day after sunset. The average time of observation each day 
being two hours, this time will be tripled on each clear day 
and night. 

" 2. Necessity for encamping at many stations may be avoid- 
ed, where now the probabilities of a long detention and the 
lack of any decent quarters within a reasonable distance require 
the transportation and use of equipage. 

"The conclusions to which the experiments and results have 
led me may be generally summed up as follows : 

" I. That night observations are a little more accurate than 
those by day, but the difference is slight so far. 

" 2. That the cost of apparatus is less than that of good 
heliotropes. 

" 3. That the apparatus can be manipulated by the same 
class of men as those whom we employ as heliotropers. 

" 4. That the average time of observing in clear weather 
may be more than doubled by observing at night, and thus 
the time of occupation of a station proportionally shortened. 
Hazy weather, when heliotropes cannot show, may be utilized 
at night. 

" 5. That reflector-lamps, or optical collimators, burning coal- 
oil, may be used to advantage on lines of 43.5 miles and under. 
But for longer lines the magnesium lights will be best and 
cheapest, as being the most certain. 

" 6. That for the present we should keep up both classes of 
observation, both by day and night ; and that the observers in 
charge of the various triangulations should be informed of 
the progress already made, and encouraged to improve on the 
methods and materials thus far employed in night observa- 
tions." 

At this time many of the parties in charge of triangulation- 



84 GEODETIC OPERATIONS. 

work, under the auspices of the Coast Survey, make night ob- 
servations. The wisdom of this plan is duly appreciated by 
all who have observed in the Eastern or Middle States. 

It might be safely said that more time is spent in waiting for 
suitable weather than in reading the angles, and any means for 
diminishing this waste will be gladly adopted, especially by 
those who have had their patience taxed by having to wait day 
after day for the haze to pass by. 

For short sights or for secondary triangulation a reflecting- 
surface, such as a tin cone, will be sufficient. Still better is a 
contrivance made of tin, in the shape of the children's toy, that 
is made to revolve by a current of air, and fixed on an axis in 
the top of a pole or tree. If it is of the proper shape, in turn- 
ing it will catch the sun's rays at the right angle to send a re- 
flection to the desired point, except when the sun is on the 
opposite side from the observer. In lines still shorter a simple 
pole, supported by a tripod, or a straight tree will answer. 
Care must be taken, however, to have the pole or tree no larger 
than is necessary to render it visible, as large bodies are diffi- 
cult to bisect. A diameter of 6 inches will subtend an angle 
of one second at a distance of 20 miles ; for 40 miles, 12.3 inches ; 
and at 60 miles, 18.5 inches. Sights have been made upon 
a tree 12 inches in diameter at a distance of 55 miles. 

Much time can be gained and accuracy secured by making 
the observations at the most favorable time. For instance, if 
a pole is to be sighted, the proper time is in the morning when 
looking towards the east, and in the evening when looking 
westward. If a reflecting object is used, the opposite rule to 
the above must be followed. 

It is frequently necessary to elevate the instrument and ob- 
server in order to obtain a longer length of line, or to overcome 
some impediment. Fig. 9 will give an idea of the form that 
has been found most convenient. When it is to be constructed 
on a hill or mountain, it will be found advisable to cut the 



FIELD-WORK OF THE TRIANGULA TION. 



%% 




Fig. 9. 



86 GEODETIC OPERATIONS. 

timbers at the bottom, in order to save the transportation of 
useless materials. 

In order to secure the requisite stability, and to prevent 
shaking of the instrument by the observers moving around, it 
is necessary to have a double structure — one for the theodolite, 
and one to support the platform for the party observing. For 
a low structure the form used by the Prussian Geodetic Insti- 
tute will be found sufficiently firm. It is a vertical piece of tim- 
ber to support the instrument, braced by a tripod, the whole sur- 
rounded by a quadrangular platform. But when a height of 
more than twenty feet is needed, the kind devised by Mr. 
Cutts, and improved by Captain Boutelle, will be found more 
satisfactory. 

I have worked on several of this pattern, and can vouch for 
their rigidity ; and when an awning is attached to the legs of 
the scaffold to shade the tripod, the unfortunate results of 
" twist " from the action of the sun's rays are avoided. 

From a glance at Fig. 9 it will be seen that the signal con- 
sists of two parts — a tripod and a square scaffold. It is the 
average experience that a safe signal, strong enough to with- 
stand the heaviest winds we have, should be built of timbers 
6 by 8 inches, with diagonal braces 2 by 2 and 3 by 3. The 
size of the base is a function of the altitude, a good ratio 
being one foot radius for every eight feet of elevation. The 
legs of the tripod should be set three feet in the ground, and 
would, if continued, meet at a point four feet above the plat- 
form. So that for a signal whose scaffold is to be eighty feet 
above the station-surface we would have eighty-seven feet for 
the vertical height of the tripod, and the radius of the base 
would be - 8 g I +o.67 ft. = 11.54 ft. 

To lay out the base, drive a stub in the ground at the cen- 
tral point, and with a radius equal to that computed describe 
a circle ; mark off on this circumference points with a chord 
equal to the radius, and the alternate points will be the places 
for the feet of the tripod. With a level, or an instrument that 



FIELD-WORK OF THE TRIANGULA TIOJV, 87 

can be used as a level, the bottom of the holes for the tripod 
can be placed on the same plane, by marking on a rod a dis- 
tance that is equal to the height of the axis of the instrument 
and three feet more, then the holes are to be dug until this 
mark coincides with the cross-wires of the telescope when the 
rod is in each hole. 

The tripod, being the highest and the innermost structure, 
should be raised first. The plans adopted for this differ with 
different persons ; some frame two legs with their bracing, raise 
them with a derrick, guy their tops, raise the third and brace 
it to the other two. A platform is built on the top of this on 
which the derrick is placed, another section is then lifted into 
place as before, the derrick again moved up until the top is 
reached. Then the blocks are attached to the top of the tripod, 
which is well guyed, and the sides of the scaffold raised as a 
whole or in sections. 

If the station is wooded, one or two large trees may be left 
standing and the blocks attached to their tops for raising the 
timbers. Signals ninety-four feet high have had their sides as 
a whole put in place, held there with guys until the opposite 
pair was raised and the whole braced together. This can also 
be done in the case of the tripod, by laying the single piece 
down with its foot near its resting-place, and the pair lying in 
the same direction framed together ; then with ropes rigged to 
a tree left standing, or to a derrick, the pair is raised until it 
stands at the right inclination, and held in place with ropes 
until the single piece is brought into position. To keep the 
feet from slipping, an inclined trench can be made towards the 
hole, or they can be tied to trees or a stake firmly driven into 
the ground. I have put up tripods in a way still different. 
By framing one pair, and attaching between their tops the top 
of the third by means of a strong bolt, the whole stretched out 
on the ground in the shape of a letter " Y," with the feet of 
the pair fastened near their final resting-place. The apex is 
lifted and propped as high as possible, then a rope is passed 



88 



GEODETIC OPERATIONS. 



through between the legs of the pair and attached to the leg of 
the single one near its lower end. It will be seen that as 
this leg is drawn towards the other two the apex is hoisted up. 

I have erected a high signal in this way by hitching a yoke 
of oxen to the single leg and hauling it towards the other two. 
If a tree should be in a suitable place, a rope passing through 
a block, attached as high up as possible in the tree, will be of 
great service in hoisting the apex. 

A good winch will be of great use, and plenty of rope will 
be needed, and marline for lashing. If all the timbers are cut 
and holes bored ready for the bolts, the labor of erection will 
be of short duration. Captain Boutelle's tables, enabling one to 
cut the timbers for a signal for any height, are inserted here : 
DIMENSIONS IN FEET. 



Tripod. 


Scaffold. 


Vertical 

height 
of floor 
above 
station 

point. 


Vert. 

length. 


Slant 
length. 


Three feet below 
station-point. 


Vert, 
length. 


Slant 
length. 


Three feet below 
station-point. 


Rad.-f 0.67 


Side of eq. 
triangle. 


One half 
diagonal. 


Side of 
square. 


32 
48 
64 
80 
96 


39 
55 
7i 
87 
103 


39-31 
55-43 
71-55 
87.68 
103.80 


5-54 

7-54 

9-54 

11.54 

13-54 


9.60 
13.06 
16.52 
19.99 
23-45 


38 
54 
70 
86 
102 


38.52 
54-75 
70.97 

87.19 
103.41 


14-33 
17.OO 
19.66 
22.32 
25.OO 


20.26 
24.04 

27.80 
31.57 
35-35 





DIMENSIONS 


OF TRIPOD. 




Slant dist. 


Vert. dist. from 


R = radius. 


Length of hor. 


Length of 


Size of 


from top. 


top = L. 


= ¥ +o.66 7 . 


brace = 1.732/?. 


braces. 


braces. 


Feet. 




Feet. 


Feet. 


Feet. 


Inches. 


O 


O.OO 


O.667 








5 


4.96 


I.287 


2.229 






8 


7-94 


I.659 


2.873 






13 


12.90 


2.279 


3-947 


6.02 


3 by 2 


20 


19.85 


3.I48 


5-452 


8-39 


3 by 2 


29 


28.78 


4.264 


7.385 


II.OO 


3 by 2 


40 


39-69 


5.628 


9.748 


13.87 


3 by 2 


53 


52.59 


7.240 


12.540 


I7-05 


3 by 3 


68 


67.48 


9.IO2 


15.765 


20.55 


3 by 3 


85 


84-35 


II. 212 


19.420 


24.40 


3 by 3 


103 . 80 


103 . OO 


13-542 


23-455 


25.OO 


3 by 3 



FIELD-WORK OF THE TRIANGULATION. 



8 9 



Ground 

Bottom of holes 


Vertical 

length 

from top. 


Slant 
length 
along 
outside 
edge. 


Slant 
length 
along 
centre 
post. 


Halfdiag. 
from 
station- 
point to 
outside 
edge. 


Hori- 
zontal 
braces 
side of 
square. 


Size of 
hori- 
zontal 
braces. 


Diag- 
onal 
braces. 

Feet. 


Size of 
diag- 
onal 

braces. 

Inches. 


Feet. 
3 
19 
35 
5i 
67 
83 
99 

102 


. Feet. 

3-05 

IQ.27 

35-49 
51.71 
67.93 
84.15 
IOO.38 
IO3.42 


Feet. 

67.50 

83.60 

99.70 

IO2.72 


Feet. 

8.49 
IT. 15 
13 82 
I6.49 

I9-J5 
21.82 
24.50 
25.00 


Feet. 

I2.00 

15.77 

19-54 

23.32 

27.08 

30.86 

34.65 

35.36 


Inches. 

3 by 4 

3 by 4 

3 by 4 

4 by 4 
4 by 4 
4 by 4 


21.27 

23.90 
26.81 
29.91 
21.66 

*22.23 


3 by 3 
3 by 3 
3 by 3 
3 by 4 
3 by 4 
3 by 4 









*One foot from ground. 



The floor of the scaffold should be twelve feet square, giving 
room for a tent large enough for the observers to move around 
in, and sufficient space outside to pass around while fastening 
the tent to the railing. 

A good shape for an observing tent is hexagonal, four and 
a half feet across, and six and a half high, one side opening for 
its entire length for exit and entrance, and the other sides hav- 
ing a flap that opens from the top to a little below the height of 
the instrument. This will keep out the sun, and also, by open- 
ing only that part that is needed, the tendency of the wind to 
cool the sides of the circle unequally can be diminished. A 
corner post will be needed at each vertex, and the top can be 
supported by a rafter running from each corner of the platform 
meeting over the centre. To determine the size of the base 
of the scaffold, we find the ratio of the half diagonal to the ver- 
tical height and add to this the half diagonal of the top. One 
foot in six has been found to give stability to the signal, so that 
for a scaffold 80 feet high with 3 feet in the ground, we have 
for the half diagonal of the base &£- -{- the half diagonal of the 
top = 23 feet, and the side of the square 27.6 feet. The slope 
can be found by trigonometry, tan. of slope = vertical height 
divided by half the difference of the upper and lower diag- 



9<3 GEODETIC OPERATIONS. 

onals. It is well to brace the signal by wire guys running from 
each length of timber in the scaffold legs. 

Probably the highest signal ever erected was built by Assist- 
ant Colonna of the U. S. Coast and Geodetic Survey in Cali- 
fornia. 

A large red-wood tree was cut off ioo feet from the ground 
and a twofold signal built, — a platform fastened to this high 
stump, and a quadripod from the ground for the support of the 
instrument. The total height was 135 feet. The observers 
were hoisted up in a chair attached to a rope passing through 
a fixed pulley at the top, and hauled by a winch on the ground. 

When the country is approximately level, the curvature of 
the earth will obstruct a long line of sight, unless the instru- 
ment be elevated or a high signal erected. When we know 
the distance within a mile or two between the points on which 
it is desired to establish stations, the problem is to find how 
high the signals or scaffolds must be in order to be intervisible. 
Also, when two suitable points of known altitudes are chosen, 
with an intervening hill of known elevation, the problem is to 
find how high one must build to see over it. 





Let h = height in feet ; 

d = distance of visibility 

to horizon in feet ; 
R = average radius of cur- 
vature in feet, 



Fig. xo. 1o S R = 7.6209807. 



The distance d being a tangent, it is a mean proportional be- 
tween the secant and the external segment, that is, h : d :: d 
: h + 2R, but h is so small compared with 2R that it can be 
omitted, and we have h=o.66?2d\ This is to be increased by 
its 0.07th part for terrestrial refraction, making h = 0.7139a? 2 , 

Vh 
or d = —z — . 
0.845 



FIELD-WORK OF THE TRIANGULATION. 9 1 

If we wish to know how far above the horizon the line of 
sight passes from two points of known elevation, we find the 
distance -to the point of tangency. 

Let D — the whole distance ; 
d — the shorter distance ; 
a — the height above the tangent ; 
m — the coefficient of d 2 in the above expression. 

h - a = md\ H-a= m(D - df = mD 2 - 2tnDd+ md 2 ; 

by subtraction 

tf—k = mD 2 — 2mDd, or 2mDd = mD 2 — (H — h)\ 

J mD 2 - ( H-h) 
therefore, d = ^ . 

This gives the distance from the lower point to the point of 
tangency ; then the height at which this tangent strikes either 
station can be found by the above formula, h — 0.711yd 2 , or 
a — h — o.yi4d 2 . 

If there is an intervening hill, we first compute the point of 
tangency of the line from the higher station ; then, how high 
up the intervening hill this tangent strikes. To this add the 
amount by which the lower hill exceeds this tangent plane : if 
this be more than the height of the intervening hill, it can be 
seen over ; if less, the difference will show how much must be 
added to the height of the terminal stations. 

If the intermediate hill be so heavily timbered as to render 
it impracticable to have it cleared, the height of the trees must 
be added to the elevation of the hill ; and at all times it is best 
that the line of sight should pass several feet above all inter- 
mediate points. The following table gives the difference be- 
tween the true and apparent level in feet at varying distances: 



9 2 



GEODETIC OPERATIONS. 





Difference in feet for — 




Difference in feet for— 


Dis- 
tance, 








Dis- 
tance, 












Curvature 






Curvature 


miles. 


Curvature. 


Refraction. 


and 
Refraction. 


miles. 


Curvature. 


Refraction. 


and 
Refraction. 


I 


0.7 


O.I 


0.6 


34 


771.3 


I08.O 


663.3 


2 


2.7 


0.4 


2-3 


35 


817.4 


114. 4 


703.O 


3 


6.0 


0.8 


5-2 


36 


864.8 


121. 1 


743-7 


4 


IO.7 


1.5 


9.2 


37 


9 J 3-5 


I27.9 


785.6 


5 


16.7 


2.3 


14.4 


38 


9 6 3-5 


134-9 


828.6 


6 


24.O 


3-4 


20.6 


39 


1014.9 


142. I 


872.8 


7 


32.7 


4.6 


28.1 


40 


1067.6 


149-5 


918. 1 


8 


42.7 


6.0 


36.7 


4i 


1121.7 


I57-0 


964.7 


9 


54-o 


7.6 


46.4 


42 


1177.0 


164.8 


IOI2.2 


IO 


66.7 


9-3 


57.4 


43 


1233-7 


172.7 


I06l.O 


ii 


80.7 


"• 3 


69.4 


44 


1291.8 


180.8 


IIII.O 


12 


96.1 


13-4 


82.7 


45 


I35I-2 


189.2 


II62.O 


13 


112. 8 


15.8 


97.0 


46 


1411.9 


197.7 


I2I4.2 


14 


130.8 


18.3 


112. 5 


47 


1474.0 


206.3 


I267.7 


15 


150. 1 


21.0 


129. 1 


48 


1537-3 


215.2 


1322 I 


16 


170.8 


23-9 


146.9 


49 


1602.0 


224.3 


1377.7 


17 


192.8 


27.0 


165.8 


50 


1668. 1 


233-5 


1434-6 


18 


216.2 


30.3 


185.9 


51 


1735.5 


243.0 


1492.5 


19 


240.9 


33-7 


207.2 


52 


1804.2 


252.6 


I55I-6 


20 


266.9 


37-4 


229.5 


53 


I874-3 


262.4 


16H.9 


21 


294-3 


41.2 


253-1 


54 


1945-7 


272.4 


1673.3 


22 


322.9 


45-2 


277.7 


55 


2018.4 


282.6 


1735-8 


23 


353-o 


49.4 


303 6 


■ 56 


2092 . 5 


292.9 


1799.6 


24 


384-3 


53-8 


330.5 


57 


2167.9 


303-5 


1864.4 


25 


417.0 


58.4 


358.6 


58 


2244.6 


314.2 


1930.4 


26 


451- 1 


63. 1 


388.0 


59 


2322.7 


325-2 


1997.5 


27 


486.4 


68.1 


418.3 


60 


2402 . 1 


336.3 


2065 . 8 


28 


523-1 


73-2 


449.9 


61 


2482.8 


347.6 


2135.2 


29 


561.2 


78.6 


482.6 


62 


2564.9 


359-1 


2205.8 


30 


600.5 


84.1 


516.4 


63 


2648.3 


370.8 


2277.5 


31 


641.2 


89.8 


551.4 


64 


2733-0 


382.6 


2350.4 


32 


683.3 


95-7 


587.6 


65 


2819. 1 


394-7 


2424.4 


33 


726.6 


101.7 


624.9 


66 


2906 . 5 


406.9 


2499.6 



The following example will illustrate its use: Suppose we 
have a line of 14 miles from A to B, and at B it is convenient 
to build a signal 21 feet high. By looking in the table in the 
fourth column, we find that the line of sight will strike the 
horizon at 6 miles, leaving 8 miles to be overcome at^4. Op- 
posite 8 in the first column we find 36.7 feet in the fourth, 
therefore at A we will have to build 37 feet to see B. 

To illustrate the second problem : 



FIELD-WORK OF THE TRIANGULATION. 93 

Let h! = height of higher station = 1220 feet; 

h = height of intervening hill = 330 feet; 
h" = height of lower station = 700 feet ; 
d = distance from h to h" = 24 miles ; 
d' — distance from h to h' = 40 miles ; 
d-\- d' — distance from h! to h" = 64 miles. 

700 feet strikes the horizon at 34.9 miles, 64 — 34.9 == 29.1 
miles from that point to the other station. By looking in the 
table at 29.1 miies, the tangent strikes the other station at 486 
feet, 1220 — 486 = 774 feet, the distance the top is above the 
tangent, and 29.1 — 24 = 5.1 miles that the point of tangency 
is from the intervening hill, and hence strikes it at 15 feet. 
Now, if we conceive a line to be drawn from the top of the 
higher to the top of the lower, we will have with the tangent 
a right-angle triangle in which the elevations at the higher 
and intervening hills above the tangent are proportional to 
their distances from the lower ; or, 24 : 64 : : x : 774, x — 290.6 ; 
that is, this sight-line strikes the intervening hill at 290 feet 
above the tangent, and the tangent strikes it at 15 feet, or the 
sight-line hits the intervening hill at 305. 6 ; as this is 330 — 
305.6 = 24.4 feet below the top, the two stations are not inter- 
visible. 

The lower station being the nearer the intervening hill, it 
would be the one to build on. To determine the height of 
the necessary signal, we have the following proportion : 

40 : 64 :: 24.4 : x, or x — 38.4 feet. 

In determining the altitude of stations, or intervening hills, 
an aneroid barometer will give a result sufficiently accurate. 
If the barometer is graduated to inches and decimals, the fol- 
lowing table, giving heights corresponding to readings of bar- 
ometer and thermometer, will be useful in estimating the 
height : 



94 



GEODETIC OPERATIONS. 



Ba- 
rom- 
eter. 




Mean 


of Observed Temperatures, Fahrenheit. 




32°. 


42°. 


52°. 


62°. 


72°. 


82 c . 


92°. 


30.O 
















29.9 


87-5 


89.4 


91.4 


93-3 


95-3 


97.2 


99.2 


29.8 


175.3 


179.2 


183. 1 


187.0 


190.9 


194.8 


198.7 


29.7 


263.4 


269.3 


275.1 


280.9 


2S6.8 


292.7 


298.5 


29.6 


351.8 


359-6 


367.4 


375-2 


383-0 


390.9 


398.7 


29-5 


440.5 


450.3 


460.0 


469.8 


479.6 


489.4 


499.2 


29.4 


52Q.5 


541-3 


553-o 


564.7 


576.5 


588.2 


600. 1 


29-3 


618.8 


632.6 


646.3 


659-9 


673.7 


687.4 


701.3 


29.2 


708.4 


724.2 


739-9 


755-4 


771.3 


787.0 


802.8 


29.1 


793.3 


816. 1 


833.8 


851.3 


869.2 


886.9 


904.7 


29.0 


888.5 


908.2 


927.9 


947-6 


967.4 


987.2 


1007.0 


28.9 


979.0 


1000.7 


1022.4 


1044 . 2 


1065.9 


1087.8 


1 109. 6 


28.8 


1069.9 


1093-5 


III7-3 


1141.1 


1164.8 


1188.8 


1212.6 


28.7 


1161.1 


1186.7 


1212.5 


1238.3 


1264. 1 


1290.0 


I3I5-9 


28.6 


1252.5 


12S0.3 


130S.1 


1335-9 


1363-8 


1391.6 


I4I9-5 


28.5 


1344-3 


1374-2 


1404.0 


1433.8 


1463.7 


1493.6 


I523-5 


28.4 


1436.4 


1468.4 


1500.2 


1532. 1 


1563.9 


1595.9 


1627.9 


28.3 


1528.5 


1562.9 


1596.8 


1630.7 


1664.5 


169S.6 


1732.7 


28.2 


1621.5 


1657-7 


1693.7 


1729.6 


1765.6 


1801.7 


1837-9 


28.1 


1714.6 


1752.8 


I790-9 


1828.9 


1867.0 


1905-2 


1943-4 


28.0 


1808. 1 


1848.3 


1888.5 


1928.6 


1968.8 


2009 . 


2049.3 


27.9 


1901.9 


1944.2 


1986.4 


2028.6 


2071.0 


2H3.2 


2T55-6 


27.8 


1996.0 


2040 . 4 


2084.7 


2128.9 


2173-5 


2217.8 


2262.3 


27.7 


2090.5 


2136.9 


2183.4 


2229.6 


2276.3 


2322.7 


2369.3 


27.6 


2185.2 


2233.8 


2282.4 


2330 7 


2379-4 


2428.0 


2476.7 


27.5 


2280.3 


2331. 1 


2381.7 


2432.2 


2482.9 


2533-6 


2584-5 


27.4 


2375-8 


2428.7 


2481.4 


2534-1 


2586.8 


2639.6 


2692.7 


27.3 


2471.6 


2526.7 


2581.3 


. 2636.2 


2691. 1 


2746.0 


2801.3 


27.2 


2567.8 


2625.0 


2681.9 


2738.9 


2795-9 


2852.9 


2910.3 


27.1 


2664 . 3 


2723.6 


27S2.6 


2841.8 


2901 .0 


2960.2 


3019.7 


27.0 


2761.2 


2822.6 


2S83.9 


2945 • 1 


3006.5 


3067.9 


3129.5 



When the station is on some hill, the name should be the 
popular designation of the elevation, or the name of the person 
who owns the property on which it is situated. 

It is a great mistake to attempt to bequeath to posterity the 
name of one of the party locating the signal. When no name 
can be found to attract attention to the locality of the station, 
a number will answer the purpose of a name. As soon as a 
station is ready for occupancy it will be found advisable to 
write a description of the signal, its position, and the way to 



FIELD-WORK OF THE TRIANGULA TION. 95 

reach it from some well-known thoroughfare, to be sent to 
headquarters. This would be of especial service in case the 
work should cease before the completion of the observations, 
to be resumed at some future time by another party. 

PORTER. 

" This point is at the head of Blue Lick, a tributary of the 
Left Fork of Twelve Pole, in Wayne County, W. Va. It is 
on the farm of Larkin Maynerd. The signal is built in the 
form of a tripod, and stands on the highest point of a large 
field. 

" A wagon can be taken up the Twelve Pole from Wayne 
C. H., and up Blue Lick to the signal. From this point can 
be seen Scaggs, Pigeon, Williamsoji, Vance, Runyan, and Rat- 
tlesnake." 

"Station No. 24: Ford County, 111.; corner of sections, 14, 
15, 22, 23 ; township, 23 ; range, 10 east." 

Each station should be provided with an underground mark, 
consisting, when accessible, of a stone pier with a hole drilled 
in the top and filled with lead bearing a cross-mark on its upper 
surface, the intersection forming the centre of the station. 
The top of the stone should not be within eighteen inches of 
the surface of the ground, so as to be below the action of 
frost, and any disturbance likely to arise from a cultivation of 
the soil. Occasionally it has been found convenient to build 
above this another pier to a height of eighteen or twenty inches 
above ground, to serve as a rest for the instrument when the 
station is occupied, and a stand for the heliotrope when the 
station is observed upon. 

When large stone cannot be had, a section of an earthen- 
ware pipe four inches in diameter may be used by filling it 
with cement and broken stone. The upper surface can be 
marked with lines before the cement sets, or a nail driven in 
while it is plastic. 



g6 GEODETIC OPERATIONS. 

When it is impracticable to dig a hole of any depth for 
a masonry superstructure on account of a stone ledge immedi- 
ately underlying the soil, it will be found sufficient to drill in 
the top of the rock a hole and fill it with lead. Whatever form 
of underground or permanent station-marks is used, it is essen- 
tial to have surface, or reference-marks. 

These are usually large stones set N., E., S., and W. of the 
centre, and at such distances that the diagonals joining those 
at the corners will intersect directly over the centre. 

For immediate use it is well to place over the centre a sur- 
face-mark, so that should anything happen to the signal before 
it is finished with, it can be replaced without disturbing the 
permanent mark. 

When the signal is a high tripod, or when it is necessary to 
raise the instrument at the time the station is occupied, the 
relative position of the centre of the station and the centre of 
the instrument must be tested at frequent intervals, as an un- 
equal settling of the signal would deflect it from or towards the 
centre. The quickest way to determine this relative position 
is to set up a small theodolite at a convenient distance from 
the centre, and fix the intersection of the cross-wires on the 
centre of the underground mark, or reliable surface-mark ; then, 
by raising the telescope, determine two points on opposite 
sides of the top of the signal. Then repeat this operation from 
a position approximately at right angles to the first position. 
Draw a string from each pair of points so fixed, and the inter- 
section of these strings will indicate the centre. If possible, 
the instrument should be placed directly over this point; if 
not, then the distance to the point, and its direction referred to 
one of the triangle-sides, should be carefully measured and re- 
corded. 

Sometimes it happens that, in the case of a very high signal 
situated on a sharp point, no position can be found from which 
both the top and the station-mark can be seen. To meet just 



FIELD-WORK OF THE TRIANGULA TION. Q? 

this difficulty, Mr. Mosman has devised an instrument with a 
vertical axis resting on levelling-screws, and so adjusted that 
when freed from errors the telescope revolves around an 
imaginary axis passing through the intersection of the cross- 
wires. 

The optical features are such as to admit of focusing it on 
objects at distances varying from a few inches to 150 or 200 
feet. To use it, you place it on a support over the centre of 
the station, the support, of course, having a hole through it. 
After levelling the instrument, move it until the cross-wires 
coincide with the station-mark ; then, by simply changing the 
focus, a point can be found in the intersection of these wires. 
This operation should be repeated with the telescope in dif- 
ferent positions; and as different points are obtained, the centre 
of the figure formed by joining these points will be the one 
desired. 

The reverse operation can also be successfully performed 
with this instrument. 

It sometimes occurs that a straight tree is used as a signal, 
in which event it is necessary to occupy an eccentric station. 
This must be marked with as much care as though it were the 
true station. The method for reducing the observed angles 
will be given in full on page 143. 

In the record-book must be kept a description of the mark- 
ings of the stations ; and when an eccentric position is occupied, 
the distance and direction already referred to are to be carefully 
entered. 

The method of observing horizontal angles must depend 
upon the accuracy desired and upon the kind of instrument 
used. Regarding the maximum error in closing primary tri- 
angles to be three seconds, or six for secondary, a number of 
precautions must be taken. The principal ones maybe classed 
under the following heads : 



98 GEODETIC OPERATIONS. 

1. Care in bisecting the object observed upon. 

2. Stability of the theodolite-support. 

3. Elimination of instrumental errors. 

4. Preservation of the horizontality of the circle. 

5. Rapidity of pointings. 

6. Observations at different times of the day. 
Conditions 1 and 2 are self-apparent, and the best means of 

compliance therewith will readily suggest themselves to the 
observer. 

The instructions for eliminating instrumental errors have 
already been given (see Chap. II.). 

When the theodolite is placed in position and levelled, see 
that the adjustments have not been disturbed before beginning 
a set of readings. If, while observing, the level shows a change 
in the horizontality of the circle, do not disturb it until the 
set is finished. But if the deflection be considerable, the read- 
ings must be thrown away. 

The advantage of pointing rapidly is the greater certainty 
of having the same state of affairs when sighting to all of the 
signals in the circuit, since it diminishes the interval during 
which there can occur unequal expansion of the circle ; twist 
in the theodolite-support, changes in the illumination of the 
different signals, or flexure of the circle from any cause. 

By making observations at different times of the day, errors 
arising from lateral reflection may be diminished because of 
the changes in the condition of the atmosphere. 

There are two principal classes of theodolites — repeating- and 
direction-instruments. The former gives a number of readings 
in a short time, but a new source of errors is introduced by the 
repeated clamping and unclamping. However, if the clamps 
do not produce what is called travelling, the principle of repeti- 
tions renders it possible to obtain a large number of readings 
on all parts of the circle, and thus tends to free the average 
from the effect of errors of graduation, for if the divisions on 



FIELD-WORK OF THE TRIANGULATION. 99 

one side of the circle are too far apart, there will be other 
parts on which the divisions are too close. In measuring an 
angle with a repeater, it is best to set the circle at zero ; 
point on the first station on the left, bisect the signal, see that 
the circle is clamped, and then turn to the next station. Read 
and record both verniers, turn the entire instrument back to 
the initial point and bisect ; then unclamp the telescope and 
point to the second station, clamp, and turn back to the first. 
Repeat this operation until the whole circle has been passed 
over; divide the last readings by the number of pointings, and 
the quotient will be the value to adopt as the average for the 
two verniers. The advantage of recording the first reading is 
that it serves as a check on the number of degrees and minutes 
in the final result. 

If there are several angles at a station, it is advisable to read 
them individually and in all combinations. Calling the angles in 
their order, I, 2, 3, 4, and 5, we read and repeat I, then 2, 
3, 4, and 5 ; afterwards 1, 2, as one angle ; then I, 2, 3, as one ; 
I, 2, 3, 4, as one ; and 1, 2, 3, 4, 5 ; also 2, 3 ; 2, 3, 4 ; and 2, 
3, 4, 5 ; then 3, 4; 3, 4, 5 ; and, finally, 4, 5, this closing the 
horizon. The advantage of this can be seen when we take up 
the adjustment of the angles around a station. When an in- 
strument can be reversed in its Y's, it will be found desirable 
to make a similar set with the telescope reversed, and record 
these as R. 

With a direction-instrument, it is not necessary to make 
these combinations. The plan s is to make 5, 7, 1 1, 13, 17, 19, 
or 23 series, by dividing the circle into such a number of parts; 
as each one is prime to two or three reading-microscopes, no 
microscope can fall on the same part of the limb twice in 
measuring the same angle. 

Suppose we decide to make eleven series, we first find the 
initial pointing for each set of the series. One eleventh of 
360 = 32 43' 38 // .2, two elevenths == 64 27' 16". 4, etc. 



100 GEODETIC OPERATIONS. 

We set the circle approximately on zero, and turn the entire 
instrument upon some arbitrary point that can be readily bi- 
sected, clamp on this, and make bisection perfect by moving 
tangent-screw of telescope if necessary. Read and record all 
the microscopes, taking both forward and backward microme- 
ter-readings as already explained. Turn the telescope until the 
next signal is bisected; read and record as before; continue 
around to the last. After recording this last reading, see if the 
signal is bisected ; if so, record the same values as just read for 
the first on the return set. When the initial point is reached, 
reverse the telescope and make a set as before, recording this set 
as R. Then set on the second position and make a complete 
set, continuing in this way until all of the positions are used. 

It is desirable to sight on all of the signals every time, and 
also on the azimuth-mark if one has been erected; but if one 
should become indistinct, while all the others show well, this 
one can be omitted and supplied afterwards. It will be seen 
at once that by this method we get an angle as the difference 
of two directions ; hence the probable error of an angle will be 
V2 times the probable error of a direction. The record-book 
should be explicit, giving the time of each pointing, position 
of circle at the initial point, the position of the telescope, D or 
R, appearance of signal (the latter is of importance in weighting 
angles); also, if a tin cone is sighted, the time and direction of 
the sun referred to the cone must be recorded as data for cor- 
recting for phase. 

If the triangulation is for general topographic purposes, it 
will be found advisable to read angles to prominent objects 
that may be in view, since if one is seen from two well-deter- 
mined stations its position can be approximately located. 
Preliminary computations should be carried along in the field, 
so as to apply reduction from eccentric stations to centre and 
deduce the probable errors of the angles. 



FIELD-WORK OF THE TRIANGULA TION. 



IOI 



If this falls beyond the predetermined limit, or if the tri- 
angles do not close after allowing for spherical excess within 
the limit prescribed, the angles should be remeasured. 

The example here given is taken from record-book just as it 
came from the field — d is the forward, and d the backward 
micrometer-reading : 



HORIZONTAL DIRECTIONS. 



Station: Holmes, W. Va. 
Observer; A. T. M. 



Instrument: 114. 



Date: Sept. 7, 1881. 
Position: 11. 



Series 
and 
No. 


Objects 
Observed. 


Time. 
h. m. 


Tel. 


Mic. 





/ 


d. 


d'. 


Mean 
d. 


Remarks. 


10 


Table Rock. . . . 


5: 10 


D 


A 
B 
C 


32 


44 


48 
25 
25 


48 
24 
25 


32.50 


Weather clear, at- 
mosphere moder- 
ately clear. 




Somerville 


5 = 8 


D 


A 
B 
C 


312 


44 


74 
53 


73 
49 
53 


58.66 


Wind S.W., light. 
Ther. 97°. 5. 




Somerville 


5: 13 


R 


A 
B 
C 


132 


44 


72 
40 
48 


71 
39 
47 


52.83 


Reversed. 




Table Rock.... 


5 : 20 


R 


A 
B 
C 


212 


44 


50 
20 
30 


50 
19 
29 


33-o 




11 


Table Rock. . . . 


5:35 


R 


A 
B 
C 


212 


44 


49 
19 
30 


48 
19 
28 


32.16 






Piney 


5 : 33 


R 


A 
B 
C 


301 


18 


64 
45 


63 
34 
43 


47-33 


Heliotrope. 
Reversed. 





In addition to the determination of the geographic positions 
of various points by triangulation, it is also possible to obtain 
with some precision their elevation, for, since we compute the 
distances between the stations, the only remaining term is the 
angle of depression or elevation from the station occupied to 
each that can be seen. This necessitates a determination, by 
levelling, of the height of the initial point only. 

The field-work can be easily described as consisting of a 



102 GEODETIC OPERATIONS. 

number of readings of the angles from the zenith to each sta- 
tion. In the computation given on page 90, it will be seen 
that vertical refraction affects this angle ; but if the zenith-dis- 
tances be measured from each station to the others at the 
same time, supposing the refraction to be equable throughout 
the intervening space, the uncertainty caused by the unknown 
deflection of the sight-line will be eliminated. 

But as it is not always feasible to have all the points oc- 
cupied at the same time, the zenith-distances can be meas- 
ured on different days, and when possible, under such varying 
atmospheric conditions as to secure the same average relative 
refraction. The best time is between the hours 10 A.M. and 
3 P.M. The height of the theodolite above ground must be 
known, as well as that of the signal sighted. 

In i860, Assistant Davidson organized a series of experi- 
ments to obtain a comparison of the various methods of deter- 
mining altitudes. He used a Stackpole level, a rod carefully 
compared with a standard and levelled in both directions. The 
measures of zenith-distances were reciprocal. They were 
made seven times daily for five days. The barometric series 
consisted of hourly readings during five days of a mercurial 
barometer, attached, detached, and wet-bulb thermometers. 
The differences in the altitudes are : 

As determined by levelling, 598.74 metres. 

" " zenith-distances, 598.64 " 

" " atmospheric pressure, 595.26 " 

REFERENCES. 

U. S. Coast and Geodetic Survey Reports, 1876, pp. 238- 
401; '80, pp. 96-109; '82, pp. 151-208. 
Puissant, Geodesie, vol. i. pp. 350-376. 
Bessel, Gradmessung in Ostpreussen, pp. 59-128. 



FIELD-WORK OF THE TRIANGULA TJON. 103 

Ordnance Survey, Account of the Principal Triangulation, 
pp. 1-61. 

Struve, Arc du Meridien, vol. i. pp. 1-35. 

Publications of the Prussian Geodetic Institute, especially 
" Das Hessische Dreiecksnetz" and " Das Rheinische Dreiecks- 
netz," II. Heft. 



104 GEODETIC OPERATIONS. 



CHAPTER V. 

THEORY OF LEAST SQUARES. 

WHEN in the various measurements of a magnitude a num- 
ber of results are obtained, it is a matter of great importance 
to know which to regard as correct. That all cannot be cor- 
rect is apparent, and that some one is true may safely be 
assumed. Errors which render a magnitude too great are 
called negative errors, and those which make it too small 
are positive errors. Should, for instance, the true length of a 
line be 73.45 chains, and its length found by measurement to 
be 73.44 chains, the error would be -f- O.OI chain ; while if 
the measurement show 73.46 chains the error would be — 0.01 
chain. 

It may be accepted as a general rule that positive and nega- 
tive errors are equally probable ; also, that small errors are 
more likely to occur than great ones, since the tendency to 
commit a great error would be readily detected before record- 
ing it, while those smaller could not be easily distinguished 
from the value afterwards found to be correct. 

Let the angle x be measured n times with equal care, so 
that in each result there is the same liability for an error to 
occur; let the individual values obtained be z\, v„ . . . v n . 
Since x is the true value, the errors will be : x — v v x — v it 
. . . x — v n \ these we will denote by dx^ dx v . . . dx n , and, 
from what has just been said, some are positive and some are 
negative. As there exists the same probability for the posi- 
tive as for the negative errors, and since the individual errors 



THEORY OF LEAST SQUARES. 105 

are nearly equal to each other, their sum will nearly amount 
to zero, and we may put, 

dx x -\- dx^ + dx % -j- dx k + • • • + dx n = o, 
or (x — v,) + (x — v 3 ) . . . + \x — v„) = o, 

whence nx = v 1 + v 2 + v 3 -f . . . + v n . Hence : 

(1) x — {v 1 + ^ 2 + v % -f~ . . . v n ) ^- «, which is simply the 
arithmetical mean of the 11 terms. This, however, gives no infor- 
mation as to the value of the errors. If for each positive error 
we had committed an equal negative error,the arithmetical mean 
would give the correct value, but this fortuitous elimination 
can only be expected in an infinite number of observations; 
even then it will not enable us to form any definite opinion as 
to the degree of accuracy attained in the individual observa- 
tions. In order to accomplish this, we must find some means 
for preventing the positive errors from destroying the negative 
ones. Gauss found the way by taking into account not the 
errors themselves, but their squares, which are positive, and 
hence cannot eliminate one another. 

For brevity we will write [v^\ for the series of terms involv- 
ing v, as v l3 v„ v s , . . . v„, and £[#„] for the sum of such a series. 
Hence we may put for the sum of the squares of the errors, 
S[d n x~Y, or S\_x — v 7 ^. The value of x will approach the 
nearest to its correct value when the arithmetical sum of the 
errors is the smallest, or when the sum of the squares of the 
errors is a minimum ; that is, when 5 [^/„^ 2 ] is a minimum. 

Let y = S [d n x 2 ] ; 

dy 
then by differentiation,-^- = 2S[d„x], 



106 GEODETIC OPERATIONS. 

As this is to be a minimum, we place the first differential co- 
efficient = o, 

or S\d n x\ = o, or 5 [(x — z>„)] = o. 

If x — v x -f- x — v 2 -f- x — v 3 . . . + x — v n = o, 



»# = Z>j + Z> a + Z>, . . . + S'w 0r ■*" = 



« 



a result identical with the one previously obtained. 

The converse can also be demonstrated; that is, the arithmet- 
ical mean gives to the square of the residuals the minimum, 

[dnX] = d x x + djc -)- d % x . . . -f- d n x 

= {* - «0 + (x-v i ) + (x-v 3 ) . .. + (x-v n ) = o; 

squaring this, 

[d n xj = (x - z/,) 2 + (* - z/ 2 ) 2 + (x - v$ ...+(*- *>») 3 
= 7Z.T 2 — 2 [>„].*■ -f [zy] 2 , 

. «. W j % CO 2 

but ^r = - — -, and ^ = — £ ; 

« n 

substituting these values, 

[^r= c -?-^+M=M'- [ -?. • © 



Suppose we now take some other value, x v so that d x x x , d^x x 
. . . d H x x represent the residuals, then 



THEORY OF LEAST SQUARES. 107 

[dnXj- = d,x' -\- djc* . - . + d n xl 

= (*> - v>y+ & - v,y • - . + '(** - v»y = o 

= tf^ 2 — 2[y„~\x l + 0„] 2 ; 
substituting in this equation the value for [z/„] 2 derived from 

(1), 



therefore, [^«^i] 2 = [^«^] 2 + »(* — -^i) 2 (2) 

Since (> — ^) 2 is always positive, [a^] 2 is greater than [d n x~] 2 ', 
that is, the square of the residuals when any value other than 
the arithmetical mean is used, is greater than when the arithmet- 
ical mean is taken. From this it is seen that the arithmetical 
mean is the most probable value ; but the correct value might 
be a little more or a little less than this mean. 

When the individual results are nearly the same, we might 
be satisfied with any one ; but when there is a great range, we 
accept the average even with some trepidation. Now, if we 
had some term that depended upon the residuals, the magni- 
tude of this term might be taken as the measure of precision : 
this is what is known as the probable error. 

The development of an expression for the probable error 
has been undertaken by many persons, and prosecuted in vari- 
ous ways, but all attaining the same end. The discussion that 
follows is taken principally from Chauvenet. 

Let us recapitulate what are known as the theorems of the 
theory of probabilities. 



108 GEODETIC OPERATIONS. 

1st. Equal positive and negative errors are equally probable, 
and, in a large series of observations, are equally frequent. 

2d. There is a limit of error which the greatest accidental 
errors do not exceed ; if / denote the absolute magnitude of 
this limit, all the positive errors will be comprised between o 
and -f- /, and all the negative errors between o and — /, so that 
the errors are contained within 2l. 

3d. Small errors are more frequent than large ones. 

So that the frequency of the error may be considered a func- 
tion of the error itself. If A be an error of a certain magnitude, 
and its frequency cpA, this function will be a maximum when 
A = o, and be o when A = ± /. If we denote the probability 
of an error A by y, we have y = cpA y an equation of a curve 
in which A is the abscissa and y the ordinate ; as A has equal 
values with contrary signs, the curve is symmetrical with re- 
spect to the axisjj/, and for y = o, A = 4; o>. 

We shall therefore consider A as a continuous variable, and 
cpA as a continuous function of it. 

If there are n errors equal to A, n' = A r . . . , and the entire 
number equal to m, the respective probabilities are 



n a, n 

mA = — , cpA' = — , etc., 



71 \ ft* . . . ffl 

and the sum <pA -4- cpA' . . . = '-- = — = 1 • 



therefore, cpA -\- cpA' -f- cpA" -f- . . . = 1. 



However, the continuity of the curve requires that the suc- 
cessive values of A shall differ from one another by an infini- 
tesimal, so that the number of values for cpA is infinite. 



THEORY OF LEAST SQUARES. IO9 

Let us take the smallest unit of magnitude in the observa- 
tions as I, then the probability of the error A maybe regarded 
as the same as the probability that the error falls between A 
and A -f- 1, and the probability of an error between A and 
A -f- i will be the sum of the probabilities of the errors A, 
A -f- 1, A -\- 2 . . . A 4- (z — 1). By making i small the proba- 
bility of each of the errors from A to A -f- i will be nearly the 
same as A, and their sum will approximate zcpA. When the 
interval between the successive errors approximates an infini- 
tesimal, the expression becomes more nearly exact, and for i 
we may put dA, and write cpA . dA as the accurate expression 
for the probability that an error falls between A, and A -J- dA. 
Hence the probability that an error falls between the limits 
-f- 00 and — 00 is the sum of the elements of the form q>A . dA, 

or the integral / cpA . dA = 1. 

Suppose the quantity M be a function of x, y, z, etc., A, A\ 
A", be the errors, and q>A, cpA', (pA", their respective probabili- 
ties. 

Since the probability of M will be the product of the proba- 
bilities of the quantities of which M is a function, we may put 
P = cpA . cpA' . cpA" ... 

From preceding principles we know that the most probable 
system of values of the unknown quantities x> f, z . . . will be 
that which makes Pa. maximum ; therefore we obtain the dif- 
ferential coefficient of P with respect to each variable and 
place it equal to o. Log P varies with P, and as P is a func- 
tion of x, y, z, . . . , the differential coefficients d P with 
respect to ,r, y, z, . . . , must separately = o ; or 



I dP I dP I dP , x 

P'-dx = °> P'^ = °' P-dz=°' (I > 

But, log P — log cpA + log cpA' -}- log cpA" ... (2) 



IIO GEODETIC OPERATIONS. 

_ , dP dcpA , dcpA' , dcpA" 
Therefore, -=• = - 1 -—. A z ~-r 7 - A z -i7 7 . . . 

Divide (3) by dx, dy, and dz, ... 



(3) 



P. dx cpAdx ' cpA'dx ' cpA"dx 
dP dcpA , a&pj' , a^J" 



P . <3^/ <pzf<^/ ' cpA'dy ' cpA"dy 
dP dcpA dcpA' dcpA" 



KA) 



P . dz cpAdz ' ^zTak ' cpA"dz 
etc., etc. ; 

since the first members are equal to zero, from (1). 

In (A) let us place for ^, cp'A. dA; for ^^, q>' A' . d'A'. 

Then they will become 

fA dA f dA f tAn dA" 

V A ^+^-dx-+V J -dx-''' = °'> 



dA f dA' n dA" 



^Tz + ^'-dz- + r' J "-d-z-- - = *•> 



dA t dA' . ...dA" 

] ~dk + <? A -Tz + *" 

etc., etc. 

If x be the correct value of M, M ', M", etc., 
A = M - x, A' = M' — x, A" = M 



KB) 



THEORY OF LEAST SQUARES. Ill 

dA dA' dA" 
from which ^ = ^ = w ... = -i. 



The first equation at (B) becomes 

cp\M - x) + cp\M' - » + cp\M" - x) . . . = o. (4) 

Now, if in this equation we suppose M' = M" . . . = M 
— mN, where m represents the number of observations, and 
since the arithmetical mean is the most probable value of x, 

x= —{M+M' + M". ..) 

=-- —\_M + (m — i) {M — mN)\ 

fflr 

since there will be m — I terms after the first, each equivalent 
to M — mN, 

or x — —(M + mM — m*N — M + mN) 



= —{mM — m*N + mN) 



= M - mN+N 
= M — [m — i)N; 

or M — x = (m — i)N, 

and M' — x = M" — x — M — mN — x\ 



112 GEODETIC OPERATIONS, 

but x = M—mN-\-N\ 

therefore 

M — mN — x — M — mN — ( M — mN ' + N) 

= M—mN-M + mN - N = — N 

Substituting these values for x in (4), we get 

cp> [( m _ i)^r] + {in - \)cp\- N) = o, 

since there are m — \ terms after the first each equal to — N. 
Transposing, 

<p'Htn - i)N] = - {in - 1) p'(- N), 

9 / [ (^- I )^ ] = _ y _ 

dividing by N y 

<p\(m - i)N] __ g>\- N) 
(in - i)N ~ - N ' 

This is a true expression for all values of m, or (m — 1), or 

N{in — 1). As the second term is not affected by changes 

in m, the expression is a constant. By putting A = {ni — l)N, 

cp'A 
we will have — -r- = a constant which is called K. 



cp'A 
If ¥-7- = K t <p r A = KA. 



THEOR Y OF LEA ST SQ UARES. 1 1 3 



dcpA 
But we supposed <p' A = — . ,r (page 1 10); 



dcpA 
therefore — - A — 7-7 = KA. 

cpA .dA 



or ^r = KA . dA. 

cpA 



Integrating this, log cpA = iKA 2 -\- log c, 
or cpA = *r^ AA , 

in which e is the Naperian base. 

Since cpA must decrease as \K increases, it must be negative. 
Placing — k* for \K, we have 

<pA = ce- h ^\ 
likewise <p A' = ce - A2A ' 2 , 

<pA" =ce -***"*, 
etc. 

We found that P = q>A~ cpA' TcpA" r~. . ; 

therefore P — c(e~ h ^ + e~ * SA ' 3 + e~ * 8A " a . . .), 



P=cKe~™ (5) 



which represents the probabilities of all errors from A to A n 
inclusive. 



114 GEODETIC OPERATIONS. 

To determine the constant c, we will take the integral on 
page 109, 



cpA % dA = 1, 



and substitute for q>A, ce~ A2Aa , 



f ce~^dA=i. 



We will write f = /t*A\ and A = % ; 



c c /*+°° 

factoring -7 , J e ~ '^ = I( 

/+00 
00 



then, since this integral is independent of the variable, we may 
also put 



/ ,+~ 
w 



due-*', 

00 

by multiplication, 

y_ w dt. due- «* + «') (6) 



THEORY OF LEAST SQUARES. 115 

If we integrate between the limits 00 and o, then between 
the limits o and — 00 their sums will be the value of the defi- 
nite integral between the limits -f- 00 and — 00. 

We shall now place u = tv, and du = tdv ; then (6) becomes 



vv 



= r r °dv.dt.te-*b + *> 

t/o t/o 

/CO •>«> 

dv / dt.te-^ + *)\ 
t/o 



v being regarded as the variable, and / the constant, 

P°° I 7t 

m* = I dv . -7 — : — =r = i(tan - « 00 — tan - x o) = - : 
t/o 2(1 + v 1 ) ^ 4 

m = / dte~* = 

t/o 2 



Without changing notation : 



m =f° dte- fl =—, 



/+CO _ 

dte-**=. Vn. 

But jle-»d t =i; 



Il6 GEODETIC OPERATIONS. 



therefore T Vn = i, c ^n = h. c = — - — 

h j^/ n 



Placing this value for c in cpA = ce ~ A2A3 , 
we have <pA = — —e -* 3A9 . 

This method of obtaining the value for the definite integral 
is taken from Laplace, Mechanique Ctleste. 

Identical results are obtained by different methods by Pois- 
son and Airy. 

The probability that the error falls between A and A -f- dA 

h 

is —p^e-^^dA. and that it falls between the limits o and a is 

Vtz 

h 

- I e-wdA, as aire; 

Y t/A = o 

Let t=.kA, then A. = T ; and a 2 — A T , then t — ah. 

h h 

Substitute these values in the last integral ; it becomes 



h {** = * . 

~~T~ I - e~ hi ^dd, as already explained. 



2 /»* 



t = ah 

e~'*dt 



after multiplying by two, since the sum of negative errors is 
considered equal to the positive errors. 

This integral has been computed for values of t. A table of 
these is given in Merriman's " Least Squares." 

From this table it is found that the error which occupies the 
middle place in the series of errors, arranged in the order of 



THEORY OF LEAST SQUARES. WJ 

their magnitude, has the same number of errors above as be- 
low ; therefore, the error satisfying this condition is that for 
which the value of the integral is \. If we designate the cor- 
responding value of t by p, we find from the same table that 



p = 0.4769. 



If r be the error in a series of observations whose precision is 

7 / P 7 P 

h, we can put p = hr, r = y, h = — . 



MEAN OF THE ERRORS. 

If we have a series of m errors A, J' . . .; a positive, and a 
negative each equal to A, or 2a in all, the probabilities are that in 

11 , m, , 2a 2a ' ^, r , 

all there will be — , — ... Thq mean of these errors, suppos- 
m m 

ing each repeated a number of times proportional to the proba- 

. 2aA + 2a' A' + 2a" A" 



bility of its occurrence, is 



tit 



2A— + 2A'— + 2A>>— 



The probability of an error A has been shown equal to cpA . dA. 
So that the above expression for an infinite number of terms 
approximates a series of terms of the form 2 A . cpA . dA. But 
on page 116 



cpA =-7=e 

\7l 



A 2 AS. 



Il8 GEODETIC OPERATIONS. 

multiplying by 2A . dA, we have 

ih. 
2AcpA . dA = -_oJ*- **VJ. 



If 7/ be the mean error, 



2h 



77= / — J^^VJ = 



Vx hVn 



p 1 r 

= — ; therefore 77 = = ;= = i.i82or; 



or f = 0.84537, since p = 0.4769 

As was stated elsewhere, it is not feasible to obtain the 
mean of the error, since the negative and positive errors being 
theoretically equal, their sum will become zero. So we take 
the sum of the squares of these errors, and the square which is 
the mean of these squares is the square of the mean error. 

That is, if s be the mean error, 



»+°° h 1 

1 1 r 



Vn 2/* 2 ' 



€ = 7-7=- = = j=r — I.4826?- ; 

r = 0.6745 £. 

This value of r is the probable error of anyone of the observed 
values of the unknown quantity, x. 



THEORY OF LEAST SQUARES. 1 19 

Let us now look for an expression for the value of the 
arithmetical mean, r n . 



Equation (5), P = cpA -f- <pA' -f cpA" . . . 

— c ( e - h*tf _|_ e - A2A'2 I e - k*A"* 

= h m n~ ± m e ~ ^ 2(A2 + A ' a + A " a • • •>. 



The most probable value of the observed quantity is that 
which makes P a maximum, or that makes A" 1 -\- A'* -[- A"* . . . 
a minimum. 

But it has been shown that the arithmetical mean renders 
the sum of the squares a minimum ; therefore P represents the 
probability of the arithmetical mean when A, A\ A" . . . repre- 
sent the residuals referred to this mean. The probability of 
any other value of x, as x -\- dx, will be 

p> — J t m n - \m e - h? { (A - d? + (A' - d)* + (A" - d)* ) 
_ J l m 7t -^ e -^{{^ - 2 {±\d + md*\ ; 
but A 2 = MS\ P' = J V n 7t -\m e -h^m^^md^ . 

\£\, being the sum of the errors, = o. 

P = h m 7t -i m e~ him ^ i 

since d = o, when x = x > 

Pi P' ". h m 7t~^ m e~ him ^ : k nt Tt~^ m e" h ' l ^ n ^^' md ^ 
P ' : P' :: i : e~ h * md * 

dividing the second ratio by the third term. 

If m = 1, P: P' :: i : e~** d \ 



120 GEODETIC OPERATIONS. 

In this single observation the probability of zero-error, as in 
the arithmetical mean, is to the probability of error, d, as 
I : e-wd* % 

As h is the measure of precision of a single observation, //' 
will be the square of this measure. 

In the expression for the error of the arithmetical mean we 
find for the square of the measure of precision of m observa- 
tions h*in\ therefore the measure of precision of the arithmetical 
mean of m quantities is h Vm. That is, the measure of pre- 
cision of the mean increases as the square root of the num- 
ber of observations. 



r = --=-, r = 0.6745*, r = .6745*, ; 



therefore 



Vm 



If v, v lt v^ ... be the observed values of a quantity whose 
mean is x> the residuals will be u, u y , u n , or x — v t x — v t 
x — v a . . . If x were the correct value, x 
would be equal to A, A', A" . . . 



and me = [J a ] = [«*] ; 

m \l m 



However, this does not consider the mean errors of the resid- 
uals. Suppose A — u — d, A' = u f — d, A" = u" — d, . . . 



THEORY OF LEAST SQUARES. 121 

[zf ] = me = (u - d)* + (»' - df + {u" - dy ... 
= O 2 ] — 2[u\d + #^ a 

= M + ;; ^ a > s ^ nce M == °- 



£ 3 

d* may be taken = f 2 = — ; 
' m 



md* = w — = « a ; 
m 



so that [z/ 2 ] = [> 2 ] + f a = me ; 

transposing w£ a — e 7 = [V] ; 

e{m - i) = [„■] ; 

M 



£' = 



m — i 1 



rwv. 

~ \J m— V 



r — 0.6745 



e 

€ n = 



V m — 1 
y ;/z(/« — 1)' 



Vm 

**.= ±0.6745^/ m{m _ iy 

To determine the probable error of the arithmetical mean, 
we find the difference between each individual result and the 
mean, square these quantities, and divide their sum by 



122 



GEODETIC OPERATIONS. 



m(m — i), where m represents the number of individual 
results. Extract the square root of this quotient, and mul- 
tiply by O.6745 ; the product will be the probable error of the 
arithmetical mean. 

The whole operation can be performed by logarithms. 

To illustrate : 



Angle. 


u. 


« 2 . 


66° — 54' — 12. "5 
13. 5 

11. 3 
16. 5 

12. 3 
15. 5 


+ Li 
+ 0.1 
+ 2.3 

— 2.9 
+ 1-3 

- i-9 


1. 21 
.01 

5-29 . 
8.41 
1.69 
3.61 


Average 13. 6 




20.22 = [u-] 



The probable error of a single determination, 



/20.22 , „ 

± 0.6745A/— = ± l"-33- 



The probable error of the arithmetical mean, 



/20.22 
= ± 0.6745^ ^ 



6X 5 



If the probable errors of the means of different sets of deter- 
minations have been found, their relative weights may be 
readily ascertained. Let k, h x , /z 2 . . . be the measures of pre- 
cision, and r, r v r 2 . . . the probable errors. 

Suppose we compare our individual observations with a fic- 
titious standard whose mean error is e t ; and the actual observa- 



THEORY OF LEAST SQUARES. 1 23 

tions with a mean error e, need w in number to reduce the mean 

£ 

error of their arithmetical mean to £ ; this gives e = —7^, or 

Vw 

ws = e*. Likewise any other set would give w^ej' 2 = e*, or 

w x e Q n = w? ; that is, w x \w\\ £ 2 : f /2 . We call w, w 1 . . . , 

etc., the weights ; they are reciprocally proportional to their 

probable errors. 

The arithmetical mean of n x observations of weight w, n 2 of 

weight zc 2 , etc., would be 



n x w l 4- ^2^2 • • • -\-n„w tl 
x = 



w l -{- w s . . \ -{- w„ ' 



\nw\ , e, f. 
or ;tf = -r-=r, and e = - = - , , 

M V{w x + w, . . . iv«) 4/[w] 

where f, is the mean error of unit weight. 

Let v x = n x — x, z> 2 == ;z 2 — x, v 3 = n z — x, 

a' 2 = v* + e * ; but w t £ n = £* = w x v? + w x e Q \ 
and e* = ze> 2 £V 2 + w ^»> etc - 

If m be the sum of such terms, 

me* = [wv*] + [w]e a = [W] -f e, a 



1 y ;« — I 



124 GEODETIC OPERATIONS. 

this, substituted in the value of e 0) gives 



/ [wv 2 ] 
" y [w](m — 



i) 



In figure- or station-adjustment, if the number of repetitions, 
or some other well-established reason, does not afford weights 
for the averages used, the reciprocals of their probable errors 
can be used. While in the development of the foregoing 
formulae there were a number of assumptions, and some ap- 
proximations to cause cautious persons to distrust the absolute 
rigor of the results, it will be apparent to all that the arithmet- 
ical mean deserves a confidence that varies with different cases. 

Suppose in measuring an angle ten results are obtained in- 
dividually differing considerably among themselves. In an- 
other measurement of the same angle ten other results are 
secured with a very small range ; now, if the average be the 
same in these two cases, the latter would be more readily ac- 
cepted, as the residuals are individually smaller. So we need 
some exponent of confidence that is a function of the residuals ; 
and if our accepted value of the probable error is not absolutely 
correct, it will afford us some information as to the agreement 
of the individual results with the arithmetical mean, and in a 
number of different determinations it gives us all the relative 
information we need. 

I shall add just here, without demonstration, other formulae 
in general use in determining probable errors: 

Probable error of a single observation, 



r = 0.6745 



y m— 1 



THEORY OF LEAST SQUARES. 1 25 

Probable error of the arithmetical mean, 



r = 0.6745 



/ M , 
y m(m — 1) ' 



If m = number of observed angles ; 

r = number of conditions in a chain ; 



probable error of an adjusted angle = \/ times prob- 
able error of an observed angle, supposing the weights ap- 
proximately equal (Walker). 

If an angle be determined by a direction-instrument, its value 
will be the difference of two directions ; so that if a is the 
probable error of a direction, a V 2 will be the probable error 
of the angle. 

If r it r 2 , r 3 . . . be the probable errors of different segments 
of a base-line, the probable error of the line as a whole, R = 

Vr.' + r.' + r.'... 

We have now shown how to determine the probable error 
of an angle or a base-line. The next subject to consider is to 
what extent these errors in an angle or in a base will affect the 
computed parts. As the errors just referred to are small in 
comparison with the magnitudes themselves, we may omit in 
all the discussions into which they enter all products, and pow- 
ers above the first. All geodetic computations are based upon 
formulae relating to triangles, so we will investigate those ex- 
pressions which are of most frequent occurrence. Denoting 
the sides of a plane triangle by a, b, and c, the corresponding 
opposite angles by A, B, and C, and the errors with which they 
may be affected by da, d&, dc, dA, dB, and dC, we can find by 



126 GEODETIC OPERATIONS. 

computation the value of any three if the values of the other 
three be known (provided one be a side). The following 
formulae are those most frequently used : 



(i) tf.sin B = b.s'm A ; 

(2) c = a. cosB-{- b.cosA; 

(3) A+B +C=i8o°. 

As da represents the correction to a, a -f- da will be the cor- 
rect value of a, or a -\- da, b-\- db, c + dc, A -\- dA, B-\-dB, 
C-\-dC, will be the true values. Substituting these in equa- 
tions (1), (2), (3), we shall have : 

(4) (a + da) sin (B + dB) = (5 + db) sin {A + dA) ; 

(5) (c + dc) = (a + da) cos (B + dB)+(b + db) cos (A + dA) ; 

(6) A + dA + B + dB + C+ dC = 180 . 

But sin (B + dB) = s'mB + dB.cosB, 

since sin dB = dB, and cos dB = 1 ; 

also sin (A + dA) = sin A -J- dA cos A ; 

cos (B + </.£) = cos B — dB sin £ ; 
cos (A + ak4) = cos A — ^4 sin A' 

Introducing these values in (4), (5), (6), and omitting all 
products of da . db, etc., we shall get 



THEORY OF LEAST SQUARES. \2J 

a . sin B -f- da . sin B -f- a . *£5 cos B = b. sin A -\- db . sin A 

-{• b . dA cos A ; 

c-\-dc — a . cos B Ar da. cos B — a . dB .sinB-{-b. cos ^ 

+ ^. cos ^4 — b. dA. sin A; 

A + B+C+dA+dB + dC=iSo°. 



Subtracting equations (i), (2), (3), from these just given, we 
obtain : 



(7) da . sin B -f- a . dB cos B = db . sin A -\-b .dA cos ^4 ; 

(8) dc — da . cos B —a . dB sin B -{- db . cos ^4 — £ . a^4 sin ^4 ; 

(9) dA+dB+dC=o. 

In a similar way, expressions can be obtained for the other 
parts, as : 

(10) dc. sin A -f- c . dA cos A = da. sin C+tf . dTC cos C; 

(11) a$. sin C + ^. dC . cos C = dfc. sin Z?-f- r . </i? . cos B; 

(12) d# = dc .cos B — c . dB sin B -\- db . cos C — b . dCsin C; 

(13) d# = da. cos C — a . dC sin C-\- dc . cos A — c .dA sin A. 



Suppose that in a triangle c, A, and i? are given, and by 
means of them the values a, b, and £7 are computed. Know- 
ing the limits of errors with which these quantities are affected, 
it is required to find the limits of the errors with which the 
computed quantities are affected ; that is, knowing dc, dA, and 



128 GEODETIC OPERATIONS. 

dB, we are to determine da, db, and dC. From equations (7) 
and (8) we have : 

(14) dC=—dA—dB; 

(1 5) da . sin B — db . sin A = — a . ^5 cos B -\-b .dA cos ^4 ; 

(16) da. cos B-\-db. cos ^4 = dc -\- a . dB sin B -\- b . d A sin A. 

Multiplying (15) by cos A, (16) by sin A and adding, we get : 

(17) da(sin A . cos B -f- cos ^4 . sin i?) = dc . sin ^ -f b . aL4 

— # .dB(cos A cos i? — sin A sin i?), 

or, da . sin (A + B) = dc .sin A + b . dA— a . dB . cos (^4 + ^), 

_ _ sin A b.dA cos(A-\-B) 

da = dc . — — 7-j — : — ^r 4- -: — jrz — n~5\ — ^ • dB— — -t~a — i — »;• 

sin (A -\- B) ' sin {A -f- ^) sin (A -f B) 

Also multiplying (15) by cosi?, (16) by sin B f and subtracting, 
we get : 

db. sin (A +B) = dc. sin B + a. dB — b.dA . cos (A + B), 

M - J s[nB a ' dB ^ /1 cos ( A + B ) 

db ~ ^sin (A + B) + sin (^4 + £) 'sin (^ + £)" 

Since sin (^4 -|- B) — sm £ an d cos (A -{- B) = — cos C, we can 
write : 

_ sin A , b.dA , T „cos C 

da = afc- — 7=, + - — 7^ + a . dB- — ^. 
sin 6 sin 6 sin 6 



THEORY OF LEAST SQUARES. 1 29 

7 sin i> a.dB .cos C 

db = dc-r—^ + -^—r + b . dA- — ~. 
sin 6 sin 6 sin 6 



And, again, since sin A : sin C::a : c, sin B : sin C :: # : <*, and 
for cos C : sin C we may put cot C, the equations then reduce 
to the following very simple form : 

y a , b.dA , __ _ 

dfo = dc — f- - — ^ + # . dB cot 6, 

£ ' sin C ' 

r & , a - dB , _ , _ _ 

f ' ant ' 

or, obtaining the relation between dfa and tf, <a# and £, 

afo dc , b.dA , T „ „ 

— = — H ^—p,+ dB . cot £7, 

a c ' a.sint ' 

db dc , a . <£# , ■ . _ 

-r = \- 1 r—^+dA . cot C. 

b c ' b . sin C ' 



Suppose ^ = 564.8, ^4 = 6i° 12' 12", # = 74 16' 30", and 

that the error in c referred to c, or -, be less than 0.0001, and 

c 

the maximum error in A and B is 1". 

• • da db _ 

It is required to compute — , -7-, and dC. 



log b = 2.8894998 log a = 2.8487359 

log dA = 4.6855749 log sin C — 9.8458288 

7.5750747 2.6945647 
2.6945647 



4.8805100 = log of second term. 
9 



130 GEODETIC OPERATIONS. 

log dB = 4.6855749 
log cot C = 0.0072518 

4.6928267 = log of third term. 

First term = o.oooiooo 

Second term = 0.0000076 

Third term = 0.0000049 

0.0001125 = error of a proportional to the 
side #. 
log a — 2.8487359 log b = 2.8894978 

log dB = 4.6855749 log sin C = 9.8458288 

7.5343108 2.7353266 

2.7353266 



4.7989842 = log of second term. 



log cot C = 0.0072518 
log dA =4.68557^9 



4.6928267 = log of third term. 



First term = 0.00010000 

Second term = 0.00000629 

Third term = 0.00000493 



0.00011122 = error of b proportional to b. 



The discussion of these formulae will develop some very in- 
teresting facts concerning the best-shaped triangles to make 
use of in prosecuting accurate geodetic work. Upon inspect- 
ing the equations it will be seen that the denominators of each 
term of the second members is sin (A-j-B), consequently 
when A -\- B is nearly 180 , or when C is very small, those 
terms involving sin C or sin [A -\- B) as a denominator will be 
made quite large, and will give to da or db a value unduly great. 



THEORY OF LEAST SQUARES. 131 

Again, the second members will have the smallest value 
when sin C has its greatest value or when C = 90 ; supposing 
that C = go°, then placing sin C = sin 90 = I, the equations 
reduce to the form 



da dc , b . . db dc a ._ 



a c 



a ' ' b ~ c ^ b 



Should dA and dB be of about the same value, and b be 
greater than a, or b : a greater than I, and a : b be less than 1, 
we will have da : a greater than db \ b; or if b is less than a 
we will have da : a less than db \ b. 

From which we can see that it will be best when a = b, 
consequently when A = B. Remembering what has just been 
said, we see that the right isosceles triangle is theoretically the 
best form of triangle to make use of. 

From a similar discussion it will be apparent that if b or a 
were the given side, the smallest error in the other quantities 
would occur when B or A = 90 . 

As all the angles cannot be each equal to 90 , the best tri- 
angle is the equilateral. A similar value can be obtained by 
direct differentiation. 



a = - — 5- . b, da — d[- — 5 . b) = d[- — 5] . b 4- db- — h. 
sin B \sin B 1 \sin Bl ' sin B 



/sin A\ _ b.cosA . sin B . dA — b . sin A cos B .dB 



b. cos A. dA £.sin^4 cos B 
sin B sin B sin B 



132 GEODETIC OPERATIONS. 

_ . b a b . cos A . dA 

Since -: — -p. == - — r, we may write for : — ~ . 

sin B sin A J sin B 



a . cos A .dA . _■ 

: -j = tf COt ^ . <&4. 

sin A 



A1 . #.sin^ cos B 

Also, since # = — : — ^-, and - — ^ = coti?, 
sin B sin B 



b sin A cos B , n _ ■ .„ 

— — 5- . - — 5 .dB = — a. cot B .dB; 
sin i* sin i> 



therefore, d# = -: — ~a$ — # . cot -5 . ^5 + a cot A .dA, 
sin i? ' 



Or, by logarithms, 

log a = log sin A + log # — log sin B ; 
differentiating, 



da cos A .dA db cos i? . ^5 
tf " sin A * b sin i? 

= cot A . dA + -t — cot B.dB; 



a 
da = a cot ^4 . dA -f- t- .db — a cot i? .*/i? ; 



# sin y4 
£ = sin £ ; 



THEORY OF LEAST SQUARES. 1 33 

hence, da = — — ^db — a cot B .dB -\- a cot ^4 . ^4. 
sin B ' 

Given in a triangle the values of c, A, and C\ required to 
compute the values of a, b, and B, and so find the limits of 
errors of the latter, supposing the errors of the former are 
known. 

From the equations already given we find : 

dB = —dA —dC; 
da . sin (A + B) = dc . sin A -f b . dA — a . dB . cos (A + B) ; 
db . sin (A + B) = <fc . sin B + a . </£- £ . <£4 . cos {A + £). 

But sin (A -\- B) = sin £7, and cos (A -\- B) — — cos C ; 

hence, afo . sin C = dc . sin A -f- £ . <aL4 -f- <z . dB . cos £T ; 
db .sin C — dc. sin B -\- a . dB -\- b . dA . cos £T ; 

_ dc . a b . dA a . dB . cos C 

da = \- - — 7=r -\ : — ^ — . 

c ' sin 6 ' sin 6 

Since dB = — dA — dC, we may write : 

da dc b . dA dA . cos C dC . cos C 
a " c * a . sin C sin C sin C ' 

afc . (b — a cos QaL4 , ^ t ^ 

= h ~ • — 7^ dC. cot c ; 

£ ' a . sin 6 

since b = a cos £T -f- £ cos ^4, we may put £ cos A for b — a cos (7, 

., da dc t C . cos ^4 . aL4 _ _ _ 

then — = L- . - — aT£\cot £7; 

a c ' # . sin 6 

but # . sin C= C. sin ^4, 

therefore — = h dA . cot A — dC . cot C. 

a c 



134 GEODETIC OPERATIONS. 

Likewise, 



db 


dc . 


sin B , a.dB | dA. cos C 


b 


~ b 


sin C b sin C sin C * 




dc 
c 


a.dA a.dC dAcosC 
b sin C bs'mC ' sin C * 




dc 
c 


(b cos C — #)aL4 # . dC 
■" £ . sin (7 £ sin C 




dc 


^ . cos B .dA a . */C 




c 


£ sin C £ . sin (7 




dc 
c 


. . l _ a.dC 




b . sin c 



Let <: = 450, ^ = 53 19' 16", <7 = 6i° 42' 32", we will find 
by computation B = 64 58' 12", # = 409.855, b = 463.05. 

Suppose that c be reliable to within 0.0001 of its entire 
length, so that dc -f- c = 0.0001, and let dA—dC= 5" : 

log dA == 5.3845449 log <#:= 5.3845449 

log cot A = 9.8720420 log cot C = 9.7309796 

<£4 . cot ^4 = 0.00001805 
dc -±- c = 0.000 1 0000 
d^.cotdT = 0.000013047 



da-— a = 0.000131097 
log ^4 = 5-3845449 
log cot B = 9.6692660 

\ogdA .cotB = 5.0538109 = 0.000011319 
log <z = 2.6126301 
log <#7 = 5»3845449 

log(«.flTO = 7.9971750 

log £ = 2.66$62y6 

log sin (7 = 9.9447545 

log (b .sin C) = 2.6103821, \og(a ,dC-~b . sin C) = 5.3867929 



THEORY OF LEAST SQUARES. 1 35 

dA .cotB = 0.000011319 
a . dC -5- b . sin C — 0.000024366 

Subtracting the sum of these two quantities from dc -s- c, 
we get 

db-r-b = 0.000064315, dB = - ($" + Si = - io /r - 

Since </£T may be either positive or negative, we may select 
the sign which will give the maximum value for the error 
da -f- a. We have therefore added the expression dC. cot C. 

In the following discussion, we will do the same, suppos- 
ing the errors committed in measuring A and B were nearly 
equal and of the same sign. 

When they are nearly equal and of the same sign, according 
to the equation already given, they will compensate one 
another. But should dc have the same sign as dA and dC, it 
would lessen the value of da when we have dC . cot C greater 
than dA . cot A, since the amount of error committed in measur- 
ing the angles is less than that of the measured side. 

The value, therefore, of da -— a will become the least when 
C is less than A and acute. But if A be obtuse, consequently 
cot A negative (assuming dA and dC to be positive), da -— a 
will become the least when C is acute, for then the last two 
terms are to be subtracted from dc -5- c. 

For A = 90 , the third term will disappear entirely, 
which circumstance will be advantageous with respect to da 
-7- a. 

In regard to the side a, A must, therefore, be either a right 
angle or obtuse, and C as small as possible. 

The same reasonings apply with respect to b } with the ad- 
ditional circumstance that B should be also very small. 

Therefore, in the present instance, a large value for A and 
small values for B and C will produce the least errors. 



I36 GEODETIC OPERATIONS. 

Suppose we have the two sides and the included angle with 
their limiting errors .given, and wish to find the limiting errors 
of the unknown or computed parts ; that, is having a, b, and C, 
to compute the value of the errors in c, A, and B. 

From equations (7), (8), and (9) we have 

(1) dA+dB = -dC; 

(2) a . dB . cos B — b .dA . cos A = db . sin A — da . sin B ; 

(3) dc -f- a . dB . sin B -\- b . dA . sin A = db . cos A -\- da . cos i?. 

Substituting for dA, — (dC-\- dB), (2) becomes 
(4)a . dB. cos B -{-b. cos A (dB -f dC)=—da . sin i? -\-db .sin A. 
By expanding and transposing we get 

(5) (a .cosB-\-b. cos A)dB = — da.s'mB -\- db. sin A 

— b .dC. cos A. 

Putting c for a . cos i? -f- b . cos ^4, 

(6) c .dB = — da. sin i? + ^ • sm ^ — b.dC. cos A, 
likewise, 

(7) <: . dA = dA .s'mB— db.slnA — a.dC . cos B. 
By transposing (3), we have 

(8) dc = da. cos B-\-db . cos A — a. dB . sin B — b . dA .sm A. 



THEORY OF LEAST SQUARES. 1 37 

Multiplying this through by c and substituting for c.dA and 
c .dB, the values given on page 136, we get 

(9) c .dc = c.da . cos B -\- c.db . cos A 

— a . sin B(— da . sin B + db . sin A —b. dC . cos A) 

— b.sinA (da .sin B — db .sin A — a . dC . cos -5). 
= c .da . cos i? + c • ^ cos ^ 

-f- (# . sin B — b sin A)(da .sinB — db sin ^4) 
-f- a . b . dC(cos A.sinB-\-sinA. cos B). 

The first factor = o, and the last is sin (A -f- B) or sin £*; 

therefore 

(10) c . dc = c . da . cos B -\- c . db . cos ^ -f- <z . 3 . dc . sin C. 
By expansion and substitution, (9) becomes 

(10) dc = da . cos B -\- db . cos ^ -f- # . # . dC . sin £7 -=- c ; 
Dividing (6) and (7) by r, we get 

(11) dB = db. sin A + c — da .sin B -±- c — b . dC .cos A -f- c; 

(12) dA = — db . sin A -+- c -\- da . sin B -+■ c — a . dC . cos B -^c. 

In computation this formula is used like those already illus- 
trated, so it will be needless to give a solution of an example 
of this kind. 

The only remaining case is when we have the three sides 
with their limiting errors to find the limiting errors of the com- 
puted angles. The discussion of this problem is of interest 
simply from a theoretical point of view, since such a case will 
never arise in any one's experience. 

Rewriting equations already deduced (page 127), we start with 

(1) b . sin A .dA -\- a. sin B . dB = da . cos B -{-db . cos A —dc ; 

(2) b . cos A . dA — a . cos B . dB = da .sin B — db . sin A ; 
(3,) dA + dB + dC=o. 



I38 GEODETIC OPERATIONS. 

Solving the first two equations with reference to dA and dB } 
and substituting in the results the values of dA and dB in (3), 
and putting sin C for sin {A -\- B), — cos C for cos (A -\- B), 
we will obtain 

(4) dA = da -f- b .sin C — db . cos C -f- b . sin £* — dc . cos i? 

-f- #. sin C; 

(5) dB =z — da . cos C -7- a . sin £7 -f- db ~ a , sin C — dfc . cos A 

-r- # . sin C ; 

(6) dfC = afo(# . cos C — a)-r- a . b . sin C -f- *#(# • cos C — b) 

-7- a . b .s'mC-{- dc(a . cos ,5 -f- b . cos ^4) -r- # . £ . sin C. 

Should we have da -t- a = db -?- b = dc -r- c, the errors of the 
sides would be proportional to the sides themselves. 

The defective triangle would then be similar to the true 
triangle, and the corresponding angles would be equal each to 
each, and we would have dA = dB "= dC = o. 

When the three angles of a triangle are observed, the differ- 
ence between their sum, after subtracting the spherical excess, 
and 180 is the total error of the triangle. 

Let us call this E, the errors of the individual angles x, y, 
and z, with respective weights, u, v, w, 

x -\-y-\- z = E* 

By preceding theory ax* -\- vy* -f- wz* = a minimum ; differ- 
entiating with respect to x, y, and z, we get ax = vy = wz. 

vy vy - . vy . . vy 

x — --, z = — ; therefore, \-y-\ = E ; 

a w a ' ^ ' w 

wvy -f- way -f- uvy = waE ; 

wuE . wvE avE 

-, also x = ; 1 , z = 



wv -f- wu -\- av wv -f- wu -f- uv wv-\-wu-\-uv 



THEORY OF LEAST SQUARES. 1 39 

The limits of errors may be found in a similar manner foi 
all combinations of triangles ; hence a polygon may be decom- 
posed into triangles and the limits of error found by the 
methods just described. 

This method is not altogether satisfactory, since in the com- 
putation of the error in each triangle we use the errors of only 
two of the angles, ignoring the third. 

in A 



t- 1 sin ^4 _ 

From trigonometry, we have a = - — ~ . o\ 



by differentiation, we have 

da = a cot A .dA — a cot B.dB, . . . . (1) 

b being a constant. 

Let a, /?, and y be the measured angles, and A, B, and 
C the correct values. The triangle error, after correcting for 
spherical excess, is a-\- ft -f- y — - 180 , one third of which may 

be attributed to each angle, so the error in a = , 

and the correct value 



A -a a + fi + Y ~ 1 8 °° - 2a "" fi ~ Y + X 8 °° - (2) 

also, l ? = /> - g +^-^J^ fl y+ !Jg; ; (3) 

ar + /?+ r -l8o° 2 y-P-a-\-iSo° /N 

C — y _ _ -_ m ^ 



As A and B depend upon a, /3, and y, the total error in the 
side a, or e a will be a function of a, j3, and y; 



140 GEODETIC OPERATIONS. 

- -• = ©'<•• +(|)V +$)V. (5) 

In which e a , fy, and £ 7 are the probable errors in a, /?, and y, 

and in (i) da = a cot ^4 . <£4 — a . cot i? . dB., 

so we must obtain dA from (2), in terms of a, p, and y. 
Likewise, dB from (3), or we may write (1) 



— acotB.dl — J — L - 

V 3 



)...'. (6) 



Differentiating (6) with respect to a, ft, and y, we have 

da 2 . , I _ /2 . , I \ ' , 

-j- = -a . cot A + -a . cot B = a[- cot A 4- cot ^ ; (7) 
da 3 3 V 3 3 / 

-a cot -4 — -tf .cot.£=tf(— *cot^ — -cot B);(S) 



dP 3 3 v 3 3 

-r- = — —a cot A -\- -a cot B = a\ — -cot A -\ — 
dy 3 3 v 3 '3 



Squaring (7), (8), and (9), 

(da V 4 ^ cQta 4 ^ 2 ^ I^ 5 cot _ 2 ^ 

War/ 9 '9 '9 

feY= -^ cot 2 A + -tf 2 cot ^ . cot B + -« 2 cot 2 B 
\lfSi 9 '9 '9 

(~ Y= -^ a cot 2 A - -a* cot ^ . cot B + -a 2 cot 2 B 
\dy / 9 9 9 



THEORY OF LEAST SQUARES. 141 

therefore, supposing e a = 6p = s y = e, we have 



'••=(£)'+ ($)+(fy 

= « 2 [f a 2 cot 2 A + fa 2 cot /I . cot B + %a 7 cot 2 B\ 
= -|£V(cot 2 A + cot A . cot B + cot 2 £) ; 



f a = f^ ^(cot 3 ^4 + cot A . cot B + cot 2 ^). 

As e a is small, the second member can be converted into a 
linear unit by writing it equal to its algebraic value times sin 
1". 

£ a = e a . sin \" 4/fCcot 2 A -f cot A . cot B + cot 2 ^). (10) 

This is a rigorous expression for the probable error of a side, 
as computed from a base supposed to be free from error. 
The side a of the first triangle may be regarded as the accu- 
rate value of the base of the next triangle, and the probable 
error of another side computed, and so on through the entire 
chain. So we may put for e„, the error of the last side, 



£„ = ea sin 1" ^f^cot 2 A + cot A . cot B + cot 2 B). (11) 

In which 2 represents the sum. In determining the angles to 
be used in this formula, it must be remembered that A is op- 
posite the side whose error is being determined, and B is op- 
posite the side whose error was last computed. 




142 GEODETIC OPERATIONS. 

To illustrate : 

Starting with the base b, we first pass to u ; 

in this case A is I, B, 3 ; 

X\ "^r^* tnen to x > A is 4, i?, 5 ; 

then to a, A is 7, i?, 8. 

Hence, 2 cot a ^ = cot 2 (1) + cot 2 (4) + cot 2 (7) . . . et 

In (1 1), s is the average angle-error in the chain. If the proba- 
ble error of the base be €&', this error will be carried through 
the chain without augmentation or diminution, owing to inac- 
curacies in the angles, but it will be increased in the ratio of 
the length of the computed line to tht base. In the first 

computed side a, the error from this source will be j- e$' m Sup- 
pose this be ej, then in the next triangle, if c is computed from 

c , c a , c 

a a b b 



, r c , c a f c , 1 

a, the error £/ = — . e x == — . j- . Sy = j- e& , and so on through 



n 
the entire chain; so if n be the last line, e„' = T . £ b r . The total 

o 



error, E, from both sources, will be E = ¥£„'* -f- £ „ 2 . If each 
side of each triangle has been computed by two different 
routes, the value for E must be divided by V2", or, E = 



CALCULATION OF THE TRIANGULA TION. 143 



CHAPTER VI. 

CALCULATION OF THE TRIANGULATION. 

Having assumed in the field that we had a line of known 
or approximately known length for a base-line, we measured 
the angles of all the triangles of our net a sufficient number of 
times to eliminate instrumental errors ; and now wish to com- 
pute the distances of all the stations from one another, as far 
as possible. 

When the three angles of a triangle are measured with the 
same care, it will be found that their sum will not equal 180 
-|- spherical excess, and when two individual angles are meas- 
ured separately and then as a whole, the sum of the two will 
not equal the two when treated as a single angle ; and, again, 
the sum of the angles that complete the horizon will always 
differ from 360 . 

The problem then is to find results from a number of ob- 
served values that will approach the nearest to the truth, and 
at the same time eliminate those discrepancies just referred to. 

There are two classes of conditions that should be fulfilled : 

(a) the sum of the individual angles should equal the meas- 

ured whole ; 

(b) the sum of all the angles completing the horizon should 

equal 360 . 
The operation of filling these conditions is called station-ad- 
justment. 

(c) The three angles of a triangle should equal 180° ; 

(d) The length of every side should be the same, regardless of 

the route by which it is computed. 



144 



GEODETIC OPERATIONS. 



The filling of these conditions is called figure-adjustment. 
These two adjustments are to be effected simultaneously, since 
the same quantities occur in each. 

The method of adjustment will depend on the way in which 
the angles are read ; whether with a repeating-theodolite, or 
direction-instrument. If with the former, the average value 
obtained for each angle will be the quantity that enters the; 
equations formed by the expressed conditions. 

Before writing these equations we must correct the angles 
for run of micrometers, as already explained on page 35, and 
for phase. 

The latter is the effect of sighting to the illuminated por- 
tion of a signal instead of the centre ; it is only appreciable 
when a tin cone, or some large reflecting surface, is observed 
on. This bright spot will be exactly in the line to the centre 
when the sun is directly behind the observer, and furthest 

from the centre when the sun is at an 
angle of 90 with the cone and ob- 
server. 

Let Cbe the centre of the signal and 
O the the position of the observer, the 
distance in the figure being greatly 
shortened, proportioned to the size of 
the signal. 

The rays of the sun may be re- 
garded as parallel and illuminating 
half of the signal, as ASB. Of this 
the observer sees only ASF '; this he 
bisects, sighting to D instead of C. 
This causes an error equal to the 
angle COD. ' 

Let SCG = x, EC = r, OC = D, 
6. KF is the projection of the visible arc, and 




and 
CD 



COD = 



AK, being perpendicular to EF, and FAE a 



CALCULATION OF THE TRIANGULA TION. 145 

right triangle, AK 2 =EK.KF= EK. 2EC (nearly), or, 

EK = ^g-. but AK= EC. sin ACE, 
2EC 

hence, AK 2 = EC 2 . sin 2 ^ CE, 

EC 2 , sin 2 ACE EC. sin 2 ACE 



and £X 



2£T 



As ^6^ is small, we can write sin 2 A CE = 4 sin 2 \A CE, EK 
= 2r. sin 2 iACE, but ^CZi = (7CS, both being complements of 

ACG, or, ^4CZi = „r ; also £7? = ; substituting, CZ> = rsin 2 

CD y sin "ovf 
|.r. In the right triangle OCD, sin = -=r = -= — . As 6 is 

f sin 2 t>% 

small, sin 6 = 6 . sin 1", or, d = -^. — : — ,-.. 

' ' jP.sini" 



This correction is to be subtracted, when the sun is to the 
right of the observer, and added when the sun is to the left. In 
the case of independent angles, if both objects observed need a 
correction for phase, the two individual corrections are to be 
subtracted if they have opposite signs, and added when they 
have the same signs. 

In the principle of directions, each direction should be cor- 
rected for phase, using only the average direction in applying 
the correction, and at all times measuring the angle x about 
the mean time of the series of observations. 

A similar correction is to be applied when an eccentric sig- 
10 



I46 GEODETIC OPERATIONS. 

nal was sighted ; in this case it is necessary to know the per- 
pendicular distance from the centre of the signal to the line 
joining the observed and observing stations. This will form a 

T T 

right triangle in which sin 6 — ■=-. or 6 = -=: — : 77, in which 

6 = correction ; r, the perpendicular, and D, the distance be- 
tween the stations. This correction is additive when the point 
observed is within the angle formed by the centres of the two 
stations, and subtractive when it falls without. 

This is but a special case of reduction to centre, discussed 
on page 196, and like the latter can be applied later as well 
as at this point. 

With the average angles corrected we wish to find those 
values that will fulfil the required conditions and at the same 
contain the largest element of truth. We have seen that the 
arithmetical mean renders the sum of the squares of the resid- 
uals a minimum, and that the most probable value of a num- 
ber of disagreeing results is the one that makes the squares of 
the errors a minimum. So we shall now look for that most 
probable value which will fulfil the conditions. 

Suppose we have a series of a observations, giving m for the 
arithmetical mean, and a series of b, giving n for the mean ; the 
relative value of these two means would be to each other as 
a : b. 

Consequently the larger number of equally good observa- 
tions we have, the better relative value we will get for the 
mean. If, therefore, the first arithmetical mean, v lt be obtained 
from a series of a v the second, v„ from # 2 . . . the flth, v„, from 
a„, we will have for the most probable value of x, 






CALCULATION OF THE TRIANGULA TION. 147 

If an angle be measured 

10 times with the result 18 18' 12", 
8 times with the result 1 8° 1 8' 2", 
5 times with the result 18 18' 21", 
4 times with the result 1 8° 1 8' 30", 



_ io(i8°i8 / i2 // ) +8(i8°i8 / 2 // ) + 5(i8°i8 / 2i // ) + 4(i8°i8 / 30 // ) 
x -~ xo + 8+5 + 4 



On page 123 it was shown that the residuals, or individual 
errors, squared were multiplied by their weights, and these 
products summed, to give the square of the probable error of 
the observations as a whole. Then, since this probable error 
is obtained by taking that value which reduces the residuals 
squared to a minimum, the sum of the individual errors 
squared, each multiplied by its respective weight, must assume 
the form which renders it a minimum. 

By way of illustration, let us take the following example : 
Suppose A, B, and C, be three angles in a plane around a 
point as a common vertex, and amounting to 360 . Suppose 
the measured values be A y B, and C; a n b n and c, their true 
values, and a, 5, and c the errors of A, B, and C, so that we 
have A + a = a t , B -f- b — b n C + c = c/, also, #, + £, + c t 
= 360 . 

From a set of 10 measurements A = 120 15' 20" ; 
From a set of 12 measurements B = 132 16' 30" ; 
From a set of 15 measurements C = 107 28 r 19". 



I48 GEODETIC OPERATIONS. 

A+£ + C= 360 oo' 09" 
a / + b / + c t — 360 oo' 00" 

a + b + c = — oo° oo' 09" (1) 

Taking the sum of the squares of these errors, # 2 + # 2 + £ 8 , 
and multiplying each by its respective weight, 

10^+12^+15^. (2) 

From (1) c = — a —b — 9 ; 

c * = a *-\-b 2 + Si + 2ab+iSa + 18J; 
15^ = 15(81 +tf 2 + ^ + 2^+ i8#+i8£); 
1 $c* = 1 2 1 5 + 1 5# 2 + 1 5# 2 + $oab + 270a + 270^ ; 

substituting this value for 15^ in (2), we get 

25a 2 + 27^ + 30^ + 270^ + 270^+ 12 1 5. (3) 

According to principles already explained, we obtain the dif- 
ferential coefficient with respect to a and b, and place each re- 
sult equal to zero ; 

Soa + 303 + 270 = o ; 5^ + 3b + 27 == o ; 
54b + 30a + 270 = o ; la + 9b + 45 = o ; 

therefore, b + 3" = o, b = — 3", a = — 3".6; substituting in 

(l),*=-2". 4 . 

The same result may be obtained by using an indeterminate 
coefficient, and afterwards eliminating it, 

ioa 2 + I2# 2 + 15^ + 2cp(a + b + c\ 



CALCULATION OF THE TRIANGULA TION. 1 49 

we take 2cp to avoid the use of fractions. Differentiating this 
with respect to a, b, and c, and placing the results equal to 
zero, we get 

20a -f- 2cp = o or 10a -f- cp = o ; 
24^ -\- 2q) = O \2b-\-cp—Q\ 

ZOC-\-2cp=.Q> l^c + cp — O. 

Eliminating cp from two of these equations, we get 

10a — \2b — o\ 

12b — i$c = o ; 

also, a -{- b -{- c-\-g — o. 



By the simple elimination, we get 

*=-3".6, b=-l", €=-2".^ 

A = 120 15' 20" - 3 /7 .6 = 120 15' i6"4; 
i? = 132 16 30 — 3 .0 = 132 16 27 .0; 
C = 107 28 19 — 2 .4 = 107 28 16 .6 ; 

#, + b / 4- ^ = 360 oo' 00". 

The above is the simplest case in practice ; that is, when only 
one condition is to be fulfilled. Let us pass to a more com- 
plicated case, or when several conditions are to be fulfilled. 

Suppose, in Fig. 13, we have from repeated measurements 
the following results: 



(I) 


MON = 68° 37' 1" with the weight 5 ; 


(?) 


MOP = 140 2 19 with the weight 10 ; 


(3) 


NOQ = 134 15 41 with the weight 20 ; 



150 



GEODETIC OPERATIONS. 



(4) 


NOR = 2ii 


56 10 


with the weight 15 ; 


(5) 


POtf = 140 


30 40 


with the weight 12 ; 


(6) 


J/6>g = 202 


52 46 


with the weight 18 ; 


(7) 


;rap = 71 


25 38 


with the weight 16 ; 


(8) 


QOR = 77 


40 6 


with the weight 20. 




Upon inspection it will be seen that the following conditions 
should be fulfilled : 



(1) 
(2) 
(3) 
(4) 



( 2 ) - (1) = ( 7 ) 

(4) - (3) = (8) 

(5) + (7) - (4) 

(6) - (3) = (i> 



Denoting the corrections to the angles by a, b, c, 



(1) 
(2) 
(3) 
(4) 



[(2) + *]-[(0 + «] = [(7)+j\1; 
[(4)+rf]-[(3) + <l =[(8) + *]; 
[(5) + t\ + [(7) +*] = [(4) + d\ ; 
[(6) +/] - [(3) + *]= L(0 + «]• 



. . (A) 

• • (B) 



CALCULATION OF THE TH I ANGULATION. 151 

Substituting in these equations the angles designated by (1), 
(2), . . . (8), they reduce to 



(1) b-a-g=-2o"\ 

(2) d—c—h = 23 ; 

(3) e+g-d= 8 ; 

(4) f-c-a= 4 . 



(C) 



These are the relations that must exist between the correc- 
tions that are to be applied to the different angles. 

Squaring each symbolic correction and multiplying each by 
its respective weight, we have 

5^ 2 + io£ 2 + 20c' + I5<T + 12* 2 + 18/ 2 + i6> 2 + 20h\ (D) 

From the equations at (C) we obtain 

(1), b=a+g— 20; (2), c = d—k — 2s; 

(3), e = d - g+ 8 ; (4), f=a + c+4 = a + d— /1-23+4. 

Substituting these in (D), 

50" + lQ (a +g - 20) 2 + 2o(d- h - 2 3 ) 2 + is^ 2 

+ i2(d-g+ 8) 2 + iS(a -\-d-h- i 9 ) 2 + io^ 2 + 2oh\ 

must be a minimum. 

The square, omitting constants, gives : 

Sa 7 + iotf 2 + log* + 2oag — 4000 — 400^ + 2od* + 20// 2 

— 40d/i — 920^+920^ + I5^ 2 + I2^ 2 + i2g* — 24^ 
+ 192^ - 192^+ i8a 2 + i8^ 2 + iW -f $6ad - z6ah 

— $6dk — 684a - 684^ + 684/2 + i6> 2 -f- 20/* 2 . 



152 



GEODETIC OPERATIONS. 



Differentiating this and placing the coefficient of each equal to 
zero, we have 

33^ + i8d + log — 1 8// = 542 ; 
1 8a -f- 6$d — I2g — 38// s= 706 ; 

S a - 6d + igg = 148 ; 

ga -f- igd — 2g/i = 401. 

The solution of these equations gives a = 6.46, d = 5.89,^" = 
7.95, and A = — 7.97, which values substituted in C, give b = 



- 5-59, 



9.14, ^ == 5.94, and f == 1.32. Since the 



errors are to be obliterated in applying the correction, each 
correction must have the opposite sign to its error ; so that if 
the above represent the errors, they are to be applied with 
contrary signs to the respective angles, which reduce the 



angles to 



(1)= 68° 3 6'54 // .54 

(2) = 140 2 24 .59 

(3) = 134 15 5o .14 

(4) = 211 56 4 .11 

(5) = HO 30 34 .06 

(6) = 202 52 44 .68 
(7)= 7i 25 30 .05 
(8) = 77 40 13 .97. 



H.K 



B.K 



In order to furnish practice, 
the following observed angles 
are taken from the author's 
record-book. The corrected 
results are given, so that those 
adjusting them can verify their 
C.t -~ — work. This, however, can be 

FlG - I4 done by seeing if the condi- 

tions are fulfilled when the corrected values are taken. The 
weights are equal, and so can be omitted. 




CALCULATION OF THE TRIANGULATION. 



Corrected. 



Observed. 

(i) CT to BK = 36 24' 2 3 // .25 = 36 24' 22 // 75 ; 

(2) CTto HK= 49 53 49 -36 = 49 53 51 - 6l J 

(3) CT to C = 95 06 40 .80 = 95 06 39 .05 ; 

(4) BKtoHK— 13 29 31 .11 '== 13 29 28 .86; 

(5) £iTto C = 58 42 14 -55 = 58 42 16 .30. 



153 



B.B 




CT 



Fig. 15. 







Observed. 




Corrected. 




(I) 


BB to C = 


26° 44' 50" 


•57 = 


26°44 / 57 // 


'.82; 


(2) 


BB to H = 


62 55 56 


.14 = 


62 55 47 


.315; 


(3) 


BB to .&£" = 


85 08 27 


•43 = 


85 08 29 


.005 ; 


(4) 


C to 77 = 


: 36 IO 41 


•57 - 


36 10 49 


•495 ; 


(5) 


C to BK = 


58 23 31 


.86 = 


58 23 31 


.185; 


(6) 


H to BK = 


: 22 12 40 


•79 = 


22 12 41 


.69; 


(7) 


H to CT = 


53 09 11 


.98 = 


53 09 10 


.18; 


(8) 


BK to CT = 


30 56 26 


.69 = 


30 56 28 


.49. 



In a large number of condition equations the above opera- 
tion may be considered long and tedious, so that one of the 
following methods may be found preferable. 

Suppose we have, as the result of the same number of related 
quantities x, y, and z, the values N v N v and N z , giving the 
equations : 



154 



GEODETIC OPERATIONS. 



^x + b.y+c.z . .. = N x \ 

*S + b%y+W • -.• = N*\ 
a * x + Ky + c 3 z . . . = N 3 ; 
a n x + b H y + c,,? . . . = N n \ 



(A) 



in which the coefficients are known. As the number of un- 
known quantities is less than the number of equations, a direct 
solution is impossible. 

Designating the errors by u, we can write equations at (A), 



#i* + b x y + *,* —N t = u x \ 
ajc -\- b t y -f- c 2 z — N 2 = u„ 

a n x + b n y + c n z — N„ = u n . 



(B) 



By the principles already stated, the most probable values 
for these various quantities are those which render the sum of 
the squares of the errors, u? -f- n* . . . +^« 2 > a minimum. Plac- 
ing all terms but those depending upon x, equal to M l9 M s 
, . . M„, equations at (B) will take the form 



a x x -f- M x = u x ; 
a^x -f- M t = « a ; 



(C) 



Taking the sum of the squares of both members of the equa- 
tions at (C), we obtain 

(4* + M$+ {as + M 9 y . , . + (*.* + J/„) 9 = «,•+ a, 3 . . . «„■. 
Differentiating this with respect to *, and placing the first dif- 



CALCULATION OF THE TR1 ANGULATION. I 55 

ferential coefficient equal to zero, we have, after dividing by 2, 

a x (a,x + M) + afax + M 2 ) . . . + a n {a n x + M,) = o. 

From this, we see that to form the most probable value for x } 
we multiply each equation by the coefficient of x in that equa- 
tion, add these products, and place the sum equal to zero. By 
doing this with y and z we will obtain one equation for each 
unknown quantity, from which each can be found by the or- 
dinary methods of elimination. 

To illustrate : x-\-2y -f- 2z — 2 = o ; (1) 

— zx+ ?+ *+4 = o; (2) 

3*+ y — z— 3 = 0; (3) 

x — 2y -\- 2z — 8 = o (4) 

Multiply (1) by 1, x-\-2y-\-2z— 2=0; 

multiply (2) by — 2, ^x — 2y — 2z — 8 = 0; 
multiply (3) by 3, 9* + 3J — 3~ — 9 = °; 

multiply (4) by I, x — 2y -\- 2z — 8=0; 

by adding, *$* + y — z — 27 = o. . . . . (5) 

Multiply (1) by 2, 2x -f- 4y -f- 4^ — 4 = 0; 

multiply (2) by 1, — 2x + y -f- z -\- 4 = 0; 
multiply (3) by 1, ^ + y — z — 3 = 0; 

multiply (4) by — 2, — 2x + 4y — 4Z -\- 16 = o; 
by adding, x-\-ioy -j- 13 = o. . . , (6) 

Likewise by multiplying (1) by 2, (2) by I, (3) by — I, and (4) 
by 2, and adding, we get 

— *+ icxsr— 13 = (7) 



156 



GEODETIC OPERATIONS. 



H.K 



Eliminating (5), (6), and (7), which are called the normal 
equations, by the usual algebraic method, we find x = 2, y = 
— 1,5, and z= 1.5. 

To further illustrate this method, we 
will take another example : 



(1) H.K to C = 87 47' 42 // -5; 

(2) H.K to H =144 17 47.5; 

(3) C to H = 56 30 09 .0 ; 

(4) C to C.T= 148 04 22 .5; 

(5) ZT to C.T= 91 34 14 .5; 

(6) CT toH.K= 124 07 29 .5. 

In this figure there are three condi- 
tions to be fulfilled : 




C.T 



Fig. 16. 



(3) + (5) = (4), (1) + (3) = (2), and (2) + (5) + (6) = 360°. 

As the changes in these values will not likely affect any- 
thing beyond the seconds, suppose we designate the seconds 
of the angles by a, b . . . f y so that we will write the angles : 

(1)= 87 47' + «"; 
(2) =144 17 +b ; 



(6) = 124 07 +/ 



(3) + (5) = (4), (3)= 56°30'+c; 
(5)= 91 34'+'; 
(3) + (5) = 148 04'+ c + e, (4) = 148° 04' + d ; 



therefore, 



c -\- e — d. 



CALCULATION OF THE TRIANGULATION. 



157 



Also (I) + (3) = (2), (i) = 8 7 °47'+ a ; 
(3)= 56°30'+^, 
(1) + (3) = I44°i7'+ a + c, (2) = 144° 17' + »\ 



therefore, 



a -f- c = b. 



(2) + (5) + (6) = 360°, (2) = 144° i7'+ * 

(5)= 9i 34 + * 
(6)= 124 07+/ 

(2) + (5) + (6) = 359° 58' + b + e +/= 360 ; 



therefore, 

By observation, 



From condition, 



b+e+f= 120". 

a = 42".5 ; 

* = 47".5; 
c — 09" ; 

d = 22 r/ .5 ; 

* = i4".5; 

7-29-5. 

b+e+f= 120". 



Substituting in observation equations the values of */, and b 
as determined by the conditional equations, we can write 



a= 42".5; 

* + <: = 47 // .5 

< = 9" ; 

£ + *>:= 22 /; .5 

*= h".5; 

/= 29".5; 

£+^+/= 120". 



(« + c) = b; 
(c + e) = <t; 



158 GEODETIC OPERATIONS. 

From these we obtain the following normal equations : 

2a -f- c = 90 ; 

& + *+/= 120; 

a + 3c+e= 79; 

t>+c+3*+f= 157; 
^+^ + 2/= I49-5- 

Solving for a, b, c, e, and/*, by ordinary method of elimina- 
tion, we find 0=4i".i25, b — 48".875, c — 7">75> d=c + e 
= 49"-37$, e = 4i // -625, and/= 29 /; .5. 

This gives for the angles the following as the most probable 
values : 

(1)= 87°47 / 4i /, .i25; 

(2) = 144 17 48 .875; 

(3) = 56 30 7 -75 ; 

(4) = 148 04 49 .375 ; 

(5) = 9 1 34 4i .625 ; 

(6) = 124 07 29 .5. 

Observations with different weights can be adjusted by this 
method. Since we do not use in this case the sqicare of the 
error, or some quantity involving the error squared, but only 
the first power, we must therefore multiply the error, or 
quantity involving the error, by the square root of the weight. 
The weight can be determined from the probable error as ex- 
plained on page 123, if not taken directly from the number of 
measurements. 

When it is desired to make use of this method for adjusting 
observations of different weights, the outline of the method 
may be given as follows. 

For each of the observations write an observation equation. 



CALCULATION OF THE TRIANGULA TION. 1 59 

For each condition write a conditional equation. 

From the conditional equations obtain as many values as 
possible for one unknown quantity in terms of others, and sub- 
stitute in the observation equations. Multiply each observa- 
tion equation by the square root of its weight. Form the nor- 
mal equations and solve as in ordinary cases. 

While normal equations will afford an excellent solution for 
any number of observation and conditional equations, the 
labor becomes quite great when we have a large number of 
equations, or large quantities to handle. 

In such cases the method of correlatives as developed by 
Gauss will afford the readiest solution. This method pertains 
to equations of condition only, and in terms of corrections that 
are to be applied to the various quantities in order to make 
them fulfil the required conditions. 

Suppose a, fi, y . . . represent the corrections, and the con- 
ditional equations expressed in terms of these corrections with 
coefficients whose values are known, as well as the absolute 
term ; for instance, in the last example we had the condition 
(3) + (5) = (4). but in reality (3) + (5) = (4) + l", or (3) + (5) 

— (4) = 1". So if <*, /?, and y represent the corrections ap- 
plied to (3), (4), and (5), their algebraic sum should equal 

— i", to counteract the error -f- \ n \ that is, a -f- y — fi = — \" . 
In this case the known coefficients are I, and the absolute 
term — i". So that, in general, we may express the condi- 
tional equations in terms of known coefficients, and absolute 
terms, with the corrections as the unknown quantities ; as 

a x oc -f- a^/3 -f- a % y . . . = M 1 

Kn + hfi + Ky**- =M 2 

n x a + n 9 /3-\-n a y . . . = M n . 



l6o GEODETIC OPERATIONS. 

Since the most favorable results are obtained by making the 
sum of the squares of the errors a minimum, if we take a^ 
+ a *fi* + # 3 y 2 • • • a n <P* — M y and differentiate it with respect 
to each variable, and making the first differential equal to zero, 
we will have, after dividing by 2, 



a x da -\- a^d/3 -\- a z dy . . . + a n d<p = o ; 
b x doi -\- b^d/3 + b % dy . . . + b n dcp = o ; 



• (A) 



m x da-\- m^d/3 -f- m z dy . . . + m n dq> = o. - 

Also, o?-\- fi a -\- y* . . . + <p 2 = a minimum, or 

a ,dot-\- /3 .dfi-^- y . dy . . . -f- cp . d<p = o. . . (B) 

As the number of equations is less than the number of un- 
known quantities, a part, as M y can be found in terms of the 
others ; with these values substituted in equations (A), we will 
have M less than originally, and each of these may be made 
equal to zero. Chauvenet accomplishes this result in the fol- 
lowing way: multiply the first equation at (A) by k lt the 
second by k» the third by £ 3 , and the n\\\ by k n and equation (B) 
by — i ; then add these products. Now, supposing k v k^ . . . 
etc., are such that M of the differentials disappear, the final 
equation will contain M' — M (calling M' the original number) 
differentials with M' equations. Making them severally equal 
to zero, we get 

a \K + b x K + c A • • • m J*m — a = o ; 
ajz, + hh + *A • • • ™jzm — P = o; 

a„k M -f b„k„ -f- c n k n . . . m n k n — cp = o. 
Now, by multiplying the first by a v the second by # 8 , etc., and 



CALCULATION OF THE TRIANGULA TION. 



161 



adding the products, expressing the sum of like terms by 2, 
we get 

2a % k x + 2abk % . . . = a x ct + a$ . . . = M t . 

Likewise, multiplying the first equation by b xi the next by 
b % . . . b", we have 

2abk x + 2P&, ... = b l a + 6J3... = M t . 

This will give as many normal equations as there are unknown 
quantities k v k» etc.; so that we obtain a, /3, y, etc., in terms 
of k lt k„ etc. While the theory of this is quite complicated 
and involves a knowledge of differential equations, in practice 
it is exceedingly simple, as the appended example will show: 



(i) B.R to C= 75° 3 i'53 / '44; 

(2) B.R to R = 144 36 49 .01 ; 

(3) B.R toG= 239 35 03 .46 ; 

(4) C to R = 69 05 00 .57; 

(5) C to G— 164 02 51 .52 ; 

(6) R to G= 94 58 05 .44. 

The conditions to be fulfilled are : 

(i) + (4)-(2)=o; 

• (2) + (6)-( 3 )=o; 

(4) + (6)-(5)=o. 




B.R 



Fig. 17. 



However, we find that 



(I) +(4) -(2)= 


5" 


(2) + (6)-( 3 ) = - 


- 9 ".oi 


(4) + (6)-(5) = 


H"-49 



1 62 



GEODETIC OPERATIONS. 



and the corrections necessary to neutralize these errors will be 
— 5, -f-9-Oi, and — 14.49. Indicating the corrections by the 
same symbols we have used for the angles, and transposing the 
constant needed, we will write the above equations, 

(1) + (4) -(2)+ 5 =0, (a) 
(2) +(6) -(3)- 9-oi =0, (6) 
(4) + (6) -(5) +1449=0, W 



Now we rule as many vertical columns as there are con- 
ditions in this case, three, and as many horizontal ones as there 
are quantities to correct, — in this case six. 



In the first condition we have -|- (1) 
-\- (4) — (2), so we write -f- k l opposite 
1, -\- k x opposite 4, and — k x opposite 

2. 





1st. 


2d. 


3d. 


I 


4- ki 






2 


- ki 


+ h 




3 




- h 




4 


+ ki 




+ h 


5 






- k 3 


6 




+ h 


+ k 3 



The second condition has (2) -j- (6) — (3); so we put -f- k % 
opposite 2, and 6, and — k^ opposite 3. 

The third condition involves (4) + (6) — (5); so we put + k % 
opposite 4 and 6, and — k 3 opposite 5. 

The first equation of correlative is to contain the contents of 
the 1st and 4th horizontal columns, and minus the contents of 
the 2d ; this is determined by that equation having (1) -\- (4) 

-( 2 ). 



1st column contains -\- k x \ 
2d column contains + k x 
4th column contains -|~ k x 
1st correlative contains 3^ ■ 



k„ [signs changed as it is — (2)] 

h + K 



CALCULATION OF THE TRIANGULA TION. 



i6- 



In the second conditional equation, we have (2) + (6) — (3), 
so we take the contents of the 2d and 6th horizontal columns 
and the 3d with signs changed. 

2d contains — K~\" ^ 

6th contains -f" k^ -f- k 3 \ 

3d contains (sign changed) -f- £ 2 

(2) + (6) - (3) contains - k x + 2>K + K 

2d correlative equation. 

Likewise for the 3d we get k 1 -\- k^-\- 3^. 

Placing these correlatives in the equations involving the cor- 
rections, {a), (b), and (c), we get 



— ^i + 3*.+ £, — 9-oi =0, 
K+ £ a + 3^. + 1449 = o. 

By ordinary process of elimination, we find k x = 3". 37, £ 2 
6 V .87, 4 = - 8 7/ .24. 



Angle. 


1st. 


2d. 


3d. 


Correction. 


Corrected Angles. 


I 
2 

3 
4 
5 
6 


- k x 


+ ^2 
- £ 2 

+ ^2 


+ *3 
-Z'3 


+ 3-37 

- 3-37 + 6 87 
- 6.87 
+ 3.37 - 8.24 
+ 8.24 
+ 6. 87 — 8.24 


75°3i'56".8i 
144 36 52 .51 
239 34 56 .58 

69 04 55 .70 
164 02 59 .76 

94 58 04 .07 



These corrections are determined in this way : 

(1) is in the first condition and positive, so it is affected by 

(2) is in the first and second, — negative in the first, and posi- 

tive in the second; therefore it is affected by — k x and 
4~&: 



164 GEODETIC OPERATIONS. 

(3) is negative in the second so it is corrected by — k 2 ; 

(4) is positive in the first and third, so its correction will be 

(5) is negative in the third, therefore its correction is — k % ; 

(6) is positive in the second and third, so it will be corrected 

by + *, + *,. 

These values of k v k„ and k z , applied as just indicated to the 
observed angles, will give the most probable values for the 
angles that will make them conformable to the conditions. 

It may be noticed that the method of forming the equa- 
tions of correlatives is the same as forming normal equations. 
To illustrate, let us take (a) of the conditional equations ; the 
coefficient of (1) is -f- 1, of (4) is -|- 1, and of (2) is — I. 

Multiply horizontal column I by -f- I, = -\-k x ; 

multiply horizontal column 4 by -|- 1, = -f- k x -J- k % \ 

multiply horizontal column 2 by — 1, = ~f"^i — k 2 ; 

(1) + ( 4 ) -(2) = 3*.-*.+*; 

therefore 3^ — £ 3 -f- k z -f- 5 = o. 

This is the better plan when the coefficients are not unities. 
When the observations have different weights, the operation 

is somewhat complicated and can 
be best explained by solving an 
example ; 

(1) Cto P = 107 53' oo".07 weight 5; 

(2) Cto A — 171 42 02 .18 weight 4; 

(3) Cto B — 198 10 28 .22 weight 6; 

(4) P to A = 63 49 05 .86 weight 2; 

(5) 7*10./?= 90 17 16 .02 weight 3; 

(6) A to B — 26 28 04 .54 weight 1. 




CALCULATION OF THE TRIANGULATION. 

The conditional equations are : 



i6- 



(i) + ( 4 )-(2)+ 375=o; (a) 

(i) + (5>- (3) -12.13 = 0; (b) 

(4) +(6) -(5)- 5-62 = 0; » 

designating the corrections by the same symbols as the angles. 
If the equations on page 159 had been weighted before dif- 
ferentiation, « 2 , /? 2 . . . cp" 2 would have been multiplied by the 
respective weights of the observation to which they were to 
form corrections. These weights, say w v w 2 . . . w m being con- 
stant factors, would remain in the differentials ; so that the 
equations just referred to would have for their last term — zv J a y 
— w$ ... — w n q). Then afterwards, when multiplied by a if a„ 
etc., before summing the products, in order to get a, /3 . . . <p 
freed from factors, since we only know the values of these 
errors unaffected by their weights, we must divide the first 
equation by w v second by w„ etc. 



We make the arrangement as though 
there were no weights so far as the 
position and signs of the correlatives 
are concerned, but take the reciprocal 
of the weight of the angle as the co- 
efficient. 



Angle. 


*,. 


k 9 . 


h. 


I 


+ l*i 


+ i** 




2 


-i*i 






3 




-**» 




4 


+ Ui 




+ u 3 


5 




+ i^ 


-£&, 


6 






+ i^3 



To illustrate : the first condition equation involves a correc- 
tion to be applied positively to (1) and (4), and negatively to 
(2). And since the reliability of these angles is proportional 
to 5, 2, and 4, it is apparent that the corrections they should 
receive would be in the inverse proportion,* or \, i, and J. 



1 66 GEODETIC OPERATIONS. 

Therefore for the correction, this equation suggests that for 
(i) we should write \k x \ for (4), \k x \ and for — (2), — \k x . 

Second conditional equation involves corrections to (i) + 
(5) — (3). These angles are in point of accuracy proportional to 
their weights, 5, 3> an d 6; therefore the corrections will have 
the inverse proportion, -|-, J, and \. So we write the correc- 
tions ; the second conditional equation suggests for (1), -\-\k„ 
for (5), +# 2 ; and for - (3), - \k v 

Likewise in the third, for (4), -f- \k*\ for (6), -f \k % \ and for 

-(5), -\K 

Now, to form the equations, the first condition requires the 

sum of the quantities in the first and fourth horizontal col- 
umn, and the negative of the second. 

(1) contains \k x + \k % 

(4) contains \k x +i& 

— (2) contains \k x 

(1)+ (4) -(2) contains (i + i + i)k x +#, + *£, = 

The second condition requires (1) +(5) — (3). 

(1) contains \k x + \k, 

(5) contains -(- ^ 2 — \k % 

— (3) contains -f \k^ 
(0+ (5) - (3) contains \k x + (£ + i + £)£ 2 - #,. (/) 

Likewise, (4) contains \k x -f~ i& s ; 

(6) contains + k z \ 

— (5) contains — -J&, + i&a> 

(4) + (6) - (5) contains \k x - \K + ft + I + *)*.- te) 

Clearing equations (*), (/), and (g) of fractions and substi- 



CALCULATION OF THE TRIANGULATION. 



167 



tuting them for the values of the corrections in (a), (&), and (c) 
we get 

m +«*.+***.+ 375-0; . . : 



M +***.-*«*.- 12.13 = 0; 



±k —±°fc 
0^2 3 0^3 



(.&) 
(0 
(*) 



Reducing 



57^ -f I2£ 2 + 304 + 225.CO = o ; 
6^ -j- 214 — io^ 3 — 363.90 = o ; 
3 £ x - 2^+11/^3- 33.72=0. 



Eliminating by the usual process, we find that 

k 1 = - i6 ;/ .5o, k % = 28".o8, k % = I2".68. 
The plan for applying these values can be best exhibited : 



Angle. 


1st. 


2d. 


3d. 


Correction. 


Corrected Angle. 


I 
2 

3 
4 
5 
6 




i^2 


1^3 

-Pa 
£3 


- 3-3Q+ 5-6i 
-(-4.12) 

- 4.68 

- 8.25 + 6.34 
9.36 — 4.22 

12.68 


107 53' 02". 38 

171 42 06 .30 

198 IO 23 .54 

63 49 03 .95 

90 17 21 .16 

26 28 17 .22 



(1) is corrected by \k^ and \k» or — 3.30 -f- 5.61 ; 

(2) is corrected by — \k x or — J(— 16.50) =4.12, etc. 

In the case of weighted observation, the method of correla- 
tives is far the simplest. 

While station adjustment is of somewhat frequent occur- 
rence, yet the angles regarded as forming parts of a triangle 
more frequently require attention. The geometric require- 
ment that the three angles of a triangle equal 180 furnishes a 
condition to begin with ; likewise, these angles as individuals 



1 68 GEODETIC OPERATIONS. 

may form a part of a station condition. In this case — which is 
the rule — we combine station-adjustment with what is known 
as figure-adjustment; that is, bringing the angles into con- 
formity with the geometric requirements of the figure. 

The first geometric condition is that the angles of the tri- 
angles equal i8o° + spherical excess, or A -\- B -\- C — £ = 
i8o°, in which, A, B, and C are the measured angles of the 
triangle, and £ = spherical excess. To find what the errors 
are in the case of each triangle, it is necessary to determine 
the value of £. By geometry we know that the three angles of 
a spherical triangle bear the same relation to four right angles 
that its area bears to a hemisphere ; that is, e : 2zr::area : 27rr 2 , 

area . area 

e = -—-r— . € being small, em seconds = . sin I , 



r~ sin i 



As the triangle is small compared with the surface of the 

sphere, it may be regarded as equivalent to a plane triangle = 

• ^ a .b .s\r\ C . 

i>a.o . sin C, hence £ = — ; — : ,;, in which a, a, and C represent 

2 ' 2r sin i" ^ 

the two sides and included angle, and r the radius of a sphere. 

r can be considered a mean* proportional between the radius 

of curvature of a meridian and the normal of a point whose 

position is the centre of the triangle. 

( T 3 \ 

On page 207, R = 7- ■ . , >» , N = 



(i-e>sm*L)i J (i-Vsin*Z)t" ~~ 

a\\ — e*) 

N . R = 7 — . 3 rx3 ; dividing by I — e* and neglecting terms 

(1 — e 2 sin 8 Ly & ' *> s 

involving powers of e above the fourth, r 2 — 



1 + e* — 2e* sin 2 L 

a* 

Substituting this val- 



1 -\-e\i — 2sin 2 Z) I -f- e* cos 2L 

a. £sin (7(i+^ 2 cos2Z) i-4-* a cos2Z 

ue for r , e = r— ; Tf . 1 he factor ^— T . — 

2a sin 1" 2^ 2 sin 1" 

varies with 2Z, and can be computed with L as the variable 

for every 3c/ and tabulated ; calling this term n, £=a . b . sin C n. 



CALCULATION OF THE TRIANGULATION. 



169 



The most elaborate spheroidal triangle-computation for 
spherical excess shows that the result obtained by using the 
above formula will differ from the correct value, only in the 
thousandth part of a second. For preliminary field computa- 
tion the excess may be taken as \" for every 200 square kilo- 
metres, or 75.5 square miles ; and when the sides are 4 miles or 
under, it can be disregarded. The following table contains ;/ 
for L, from 24 to 53 30', based upon Clarke's spheroid. The 
table must be entered with the average latitude of the tri- 
angle approximately. 



Latitude. 


Log «. 


Latitude. 


Log n. 


Latitude. 


Log n. 


24° 00' 


1.40596 


34° OO' 


I.40509 


44° OO' 


I. 40410 


- 24 30 


92 


34 30 


05 


44 30 


OS 


25 OO 


83 


35 OO 


OO 


45 00 


OO 


25 30 


84 


35 30 


495 


45 30 


395 


26 OO 


80 


36 00 


9i 


46 00 


90 


26 30 


76 


36 30 


86 


46 30 


85 


27 OO 


72 


37 00 


81 


47 00 


80 


27 30 


63 


37 30 


76 


47 30 


75 


23 00 


64 


3S 00 


7i 


48 00 


69 


28 30 


1.40559 


38 30 


1.40466 


48 30 


1.40364 


29 OO 


55 


39 00 


61 


49 00 


59 


29 30 


5i 


39 SO 


56 


49 30 


54 


30 OO 


47 


40 00 


5i 


50 00 


49 


30 30 


42 


40 30 


46 


50 30 


44 


31 OO 


37 


41 00 


4i 


51 00 


39 


31 30 


33 


41 30 


36 


5i 30 


34 


32 OO 


2S 


42 00 


3i 


52 OO 


29 


32 30 


24 


42 30 


26 


52 30 


24 


33 00 


19 


43 00 


20 


53 00 


19 


33 30 


I.405I4 


43 30 


1. 4041 5 


53 30 


1. 403 14 



The spherical excess computed by this formula is for the 
entire triangle ; and, unless there is considerable difference in 
the lengths of the opposite sides, one third of the excess is to 
be deducted from each angle of the triangle; but this reduced 
value is used only in the triangle condition, and not in the 
station condition. 



170 



GEODETIC OPERATIONS. 



If in this figure we have measured the angles numbered and 
have the averages, which we will designate by the numbers, it 

will be seen that a great variety 
of conditions may be written. 
But upon examination it will be- 
come apparent that some of the 
angles are indirectly two or more 
times subjected to the same or 
equivalent conditions. For in- 
stance, if (3) + (7) + (9) = 180 , 
and (11) + (13) + (1) = 180 , the 
condition that (2) + (7) + (10) + 
(13) — 360 is already fulfilled. 
Also, if (1) + (3) = (2), and (2) + 
(5) = (4), the condition (1) +(3) + (5) = (4) is unnecessary; 
and, again, if (3) + (7) + (9) = 180 , (11) + (13) + (1) = 180 , 
and (14) + ( 2 ) + (6) = 180 , then (1) -f- (3) = (2) is unneces- 
sary. If we have the most probable value for (6) and (7), their 
difference will be (8) without involving (8) in the adjustment ; 
or if we have the best values for (8) and (6), their sum will 
give (7) ; or if we have (1), (2), and (4), we can find by subtrac- 
tion the most probable value for (3) and (5). 

From this we learn that it is useless to involve whole angles 
and all of their parts in different conditions. With such a fig- 
ure the following conditions would be sufficient : 




Fig. 19. 



(2) + (5) = (4); 

(3)+(7) + (9)=i8o°; 

(8) + (10) + (12) =180°; 

(2) + (7) + (10) + (13) =360°; 

(13) + (16) = (15). 



Other combinations could also be used. 



CALCULATION OF THE TRIANGULA TION. 



171 



gives us -j- c 



Since A, B, and C are fixed points, 



the distances AB and A C are constant; therefore - -^ repre- 



In such adjustments the method of correlatives should be 
used, as the labor does not increase rapidly in proportion to 
the increased number of conditions. 

The equations like those given so far are called angle or an- 
gular equations. The theorem in trigonometry that the ratio 
of sides is equal to the ratio of the sines of the opposite angles 
AB sin (3 ) 
sin (f)' 

AB 
AC 

sents a constant quantity, so that if (3) and (7) are changed at 
all, the sine of (3) and its correction 
must have the same ratio to the 
sine of (7) and its correction that sin 
(3) has to sin (7). This involves 
another condition, which will now 
be elaborated. 

From the theorem just referred 
to, we obtain the following equa- ci 
tions : Fig. 20. 




OB __ sin b 2 OA _ sin b 3 OD _ sin b , OC _ sin b x 

OA Z ~ sin a x ' OD ~ sin a 2 ' OC ~ sTrwzV OB ~ "sTruT/ 



Multiplying these equations together, member by member, we 
obtain 



OB.OA.OD. OC _ _ sin b 2 . sin b % . sin b A . sin b x 
OA . OD . OC • OB ~ sin a x . sin a 2 . sin a % . sin a i 



or, sin a x . sin a 2 . sin a % . sin a i = sin b x . sin £ 2 . sin b 3 . sin b t . 



172 GEODETIC OPERATIONS. 

But these are the values after correction, so we will put M v 
M„ M w M„ for a„ a„ a 3 , a,\ N v N„ N„ and N 4 for b lt b n , b„ 
and b t ; and denote the necessary corrections by v v v„ v 3 , v v and 
x v x % , x % , and x 4 . Substituting these values in the last equa- 
tion, we have 

sin (M x + v x ) . sin (M t + v t ) . sin (M 3 + v t ) . sin (M A + v 4 ) 

= sin (N x + x x ) . sin (iV 3 + x^ . sin (7V 3 + x s ) . sin (N t + *r 4 ); 

or, passing to logs, 

log sin (M x + v x ) + log sin (M t + v t ) + log sin (M 3 + v % ) 

+ log sin (if 4 - + vj 
= log sin (N x + x x ) + log sin (N t + *,) + log sin (7V 3 + x t ) 

+ log sin (^+4^). 

Since v x , v„ v z , v v x v x v x z , x v are very small, we may develop 
each of the above terms by Taylor's theorem, stopping with 
the first power of the correction : 

. /„,r , . „^ , (d log sin M.\ 
log sin (M x + v x ) = log sin M x + [ ^ — l -)v t ; 

log sin (AT, + z> 2 ) = log sin ^ + [ *^—L) Vv 

etc., etc.; 
log sin (#; + *,) = log sin ^ + ^ ^ ij*,, 

etc., etc.; 

in which v x , v„ z> 3 , v v x v x„ x a , x v are expressed in seconds, so 



CALCULATION OF THE TRIANGULATION. 1 73 

d log sin M x . . , 

that -7-r^ is the log difference tor one tabular unit for 

the angle M v or the tabular difference for M x ; let us call this 
difference d it d a , d z , d A , for M v M v M 3 , M A , and '6 X% tf a , d„ & v for 
N v N„ N v N A . 

Substituting these values in the last equation, we have 

log sin M x -\- d x v x -f- log sin M^ -f- d t v % -f- log sin M s -f- ^3 

+ log sin M A + <^ 4 

= log sin N x + ^1 + l°g sm ^"a + ^a + l°g sm ^ + ^3 

+ logsiniV 4 +^ 4 . 

When transposed, 

log sin M x + log sin M 2 -f- log sin «#/", -f- log sin M A 
— log sin TV, -— log sin N 2 — log sin N 3 — log sin N A 

= f 1*1 + <^a + ^3 + V* - ^1 - <*, - d % V % - d A V A 

— an equation in which the unknown quantities are the correc- 
tions to the angles, or the same quantities that are sought in 
the adjusting equations. 

This gives directly an equation of condition, for since the 
sum of the log sines of M x , M v M % , and M A should equal the 
sum of the log sines of N v N v N„ and N v the corrections d x v x 
+ ^a + d % v % -f d A v A should equal 6 x x x -f- 6^ -\- 6 2 x 3 -\- d A x A . 
But if the log sines of (M) differ from the log sines of (TV), then 
that amount of difference must be corrected in d x v x -f- d^ . . . 
etc. This is called the linear equation. 

By way of illustration, suppose we have the appended fig- 
ure with the average angles as given : 



(0 = 


50 3i'i 3 ".68; 


(2) = 


14 51 47 .88; 


(3) 


not needed ; 



174 



GEODETIC OPERATIONS. 




(4) = 


7i c 


> 46' 16' 


'.36; 


(5) = 


82 


32 49 


.52; 


(6) = 


32 


04 12 


49; 


(7) 


not needed ; 


(8) = 


30 


03 29 


•39; 


(9) = 


133 


03 52 


48; 


(io) = 


6; 


23 18 


•99; 


(ii) 


no 


t needed; 


(12) = 


57 


42 49 


.56. 


We first deduce the linear equa 


tion : 









Fig. 21. 



^^.sin(2) = /r5.sin(6); 
.##. sin (10) .= HP. sin (8) ; 
HP. sin (4) = HW. sin (12) ; 



by multiplication, 

sin (2) . sin (10) . sin (4) = sin (6) . sin (8) . sin (12). 

Writing for tabular difference £(2), #(io), etc., and [2], [10], 
etc. as the corrections for (2), (10), etc., we have 

log sin (2) -f- $(2) [2] -f log sin (10) + 3 (16) [10] 

+ log sin (4) + ^(4) [4] 
= log sin (6) + 6(6) [6] + log sin (8) + tf(8) [8] 

+ log sin (12) -J- #(12) [12]. 

From the table of logs, Ave find : 

log sin (6) = 9.72505722, d (6) = .00000336; 
log sin (8) = 9.69978200, tf (8) = 364 ; 

log sin (12) = 9.92705722, 6(12) = 133 ; 



sum 



29.35189644. 



CALCULATION OF THE TRIANGULATION. 1 75 

log sin (2) == 9.40910559, $ (2) = .00000794; 
log sin (4)= 9-977638I3, # (4) = 69; 

log sin (10) = 9.96526395, tf(io) = 87 ; 

sum = 29.35200767. 

log sin (2) -|- log sin (4) -|- log sin (10) 

= log sin (6) + log sin (8) -+- log sin (12) -f- 0.0001 1 123. 

As the corrections are to neutralize this difference, we write 

<K2)[2] + %)[ 4 ] + <?(l0)[l0] 

= <K6)[6] + e?(8)[8] + tf(i2)[l2] - 0.0001 1 123. 

Substituting for #(2), #(4), etc., their values, we have, after 
multiplying by 1000000 to avoid decimals, 

7. 94 [2] + .87[io] + .6o[ 4 ] 

= i-33[i2] + 3.64P] + 3-36[6] - 1 1 1.23. 

Transposing and passing to our usual notation, 

7.94(2) + .87(10) + .69(4) - 1.33(12) - 3.64(8) - 3-36(6) 

+ 1 1 1.23 = o. 

The angle equations are those involving the angles that 
will not be doubly adjusted. In the present case they will be, 
when expressed in terms of their corrections, 

(5) + (I)" (9) + 10.72 = O ; 

(9) + (2)+ (6)- 7.15 = 0; 

(I) + ( 4 ) + (12) +19.60=0; 

(5) + (8) + (io)- 22.10 = 0; 



ij6 



GEODETIC OPERATIONS. 



7.94(2) + .87(10) + .69(4) 



1.33(12) ~ 3^4(8) - 3.36(6) 

+ 1 1 1.23 = O. 



In this (3) is omitted, since if (2) and (12) are known, (3) can 
be found by subtraction. Likewise, (11) is the sum of (8) and 
(4); also (7), the difference between (10) and (6). So we now 
simply form the correlative equations from these five condi- 
tional equations. 





ISt. 


2d. 


3d. 


4 th. 


5th. 


I 


*. 




h 






2 




kn 






7-94^5 


4 






h 




.69^5 


5 


kx 






k. 




6 




k, 






— 3-36^6 


8 








k< 


— 3-64^5 


9 


-*! 


h 








.10 








k* 


.87^5 


12 






h 




- 1.33** 



The formation of the first four normal equations follows the 
principles repeatedly given, but as something new may appea** 
in obtaining the fifth equation, it will be formed in detail. 



7.94 times column 2 = 

.87 times column 10 = 

.69 times column 4 == 

1.33 times column 12 == 

3.64 times column 8 = 

3.36 times column 6 = 

Total, 



7.944 




+63.04364 
.874+ 75694 




.694 


+ 476i4 




■1-334 


+ 1.76894 






-3.644+13.24964 


3-364 




+ 11.28964 


4.584- 


• -644- 


-2.774+90.58474. 



Barlow's table of squares will facilitate work, as the coeffi- 
cients of the terms in the side equation are squared in finding 



CALCULATION OF THE TRIANGULATION. 



177 



the coefficient of the correlative corresponding to the equation fc 
of condition formed by the side equation. In this case, the 
fifth conditional equation is the side equation, and the coeffi- 
cients of k & in the fifth normal equation are the squares of 
7.94, etc. 

The normal equations are : 



3*- 


4+ k,+ 


K + 


10.72 = 0; 


- K+ 


iK 


+ 4-58*.- 


7.15 =0; 


+ K 


+ ih 


- .64*.+ 


19.60 = ; 


k, 


+ 


3*.- 2.77k, - 


9.29 = 0; 



4.58^ — .644 — 2.77^ + 90.58^ + 1 1 1.23 = o. 



The solution of these equations gives k 1 = — ".53, k 2 = 4^.40, 
k s =- 6". 66, k t = 1 ".94. K = ~ 1 "43- 

These values are applied to the various angles as indicated 
in the table just given. For instance, (2) is to be corrected by 
£, and 7.94 times k h . 

The best rule that can be given for the formation of side 
equations is to regard one of the ver- 
tices as the vertex of a pyramid, with 
the figure formed by the other points 
as the base, and take the product of 
the sines of the angles in one direction, 
equal to the product of the sines in the 
opposite direction. 

Take H as the vertex, and WPB as 
the base ; then, 




sin HWP. sin HPB . sin HBW 

= sin HBP. sin HPW . sin HWB. ; 



that is, sin (12) . sin (8) . sin (6) = sin (10) . sin (4) . sin (2), as was 
otherwise obtained. The angles at the point used as the ver- 
12 



178 GEODETIC OPERATIONS. 

tex are not involved in this equation, so they must be involved 
in a station adjustment, or in a triangle condition. 

If one should find it difficult to conceive a pyramid con- 
structed in this way, he can without trouble secure the side 
equation in the manner made use of on page 174, in which we 
started from HW . sin (2) = HB . sin (6). 

In the next equation obtain a value of HB. in another tri- 
angle, as HB . sin (10) = HP. sin (8) ; then in terms of HP., as 
HP. sin 4 = HW. sin 12. 

This is as far as we can go, as we have returned to the start- 
ing-point. Suppose we start from WP. 



WP. sin (11) = WB. sin (7) 

WB sin (6) = WH. sin ( 9 ) 

WH. sin (1) = WP. sin (4) 



by multiplying, sin (1 1) , sin (6) . sin (1) = sin (7) . sin (9) . sin (4). 
The same can be obtained by taking Was the vertex, and 
BHPas the base, the angles in one direction will give 

sin WPH. sin WHB. sin WBP = sin WBH. sin WHP. sin WPB. 

In writing down the equations to be used, a good plan is to 
put down the sides emanating from the pole to all the other 
points, putting the line first in the first member, and then in 
the second ; as, 

WP sin() = WB sin () 
WB. sin ( ) = WH. sin ( ) 
WH.sin() = WPsin() 

coming back to the first line used. Then we put in the angle 
that is opposite the side in the other term ; as, (11) opposite 



CALCULATION OF THE TRIANGULATION. 1 79 

WB, (7) opposite WP, in accordance with the trigonometric 
theorem. 

The following rule, so frequently quoted, is taken from 
Schott (C. S. Report, 1854). 

The only choice in selecting the station to be used as the ver- 
tex, or pole, as it is sometimes called, is to take that vertex at 
which the triangles meet which form the triangle equations of 
condition, and to avoid small angles, since the tabular differ- 
ences, being large, will give unwieldy coefficients. It is some- 
times difficult to determine the precise number of condition 
equations that can be formed. 

The least number of lines necessary to form a closed figure 
by connecting/ points is/, and gives one angular condition. 
Every additional line, which must necessarily have been ob- 
served in both directions, furnishes a condition ; hence a sys- 
tem of / lines between / points, I — p -\- 1 angle equations, 
where it must be borne in mind that each of the / lines must 
have both a forward and a backward sight. 

When, in any system, the first two points are determined in 
reference to one another by the measurement of the line join- 
ing, then the determination of the position of any additional 
station requires two sides, or necessarily two directions; hence 
in any system of triangles between/ points, we have to deter- 
mine p — 2 points, which require 2{p — 2) directions, or by 
adding the first 2p — 3. Consequently, in a system of /lines, 
/— (2/ — 3), or /— 2p -f- 3 sides are supernumerary, and give 
an equal number of side equations. 

We have, therefore, 

* 

l — p-\-i angle equations; 
/— 2p -\- 3 side equations; 
2/— 32> + 4 in all. 

It is apparent that each point may be taken as the pole, and 



1 80 GEODETIC OPERATIONS. 

as many side equations formed as there are vertices. In a 
quadrilateral, for instance, if four side equations are formed, the 
fourth equation would involve the identical corrections con- 
tained in the others. Since there are only 12 angles in all, 
these can be incorporated in two equations, each of which con- 
tains 6 angle corrections. 

From the formulae just given, it will be seen that 4 condi- 
tional equations will be sufficient in a quadrilateral ; 1 side 
and 3 angle equations, or 2 side and 2 angle equations, but 
never more than 2 side equations. 

The method of station adjustment differs somewhat from 
the foregoing when the values of the angles depend upon di- 
rections. 

In nearly all refined geodetic work angles are so determined ; 
that is, the zero of the circle is set at any position, the tele- 
scope is pointed upon the first signal to the left, and the mi- 
crometers or verniers read ; the telescope is then pointed to 
each in succession and the readings recorded. After reading 
the circle at the last pointing, this signal is again bisected and 
readings made, likewise with the others in the reverse order. 
The telescope is reversed in its Y's and a similar forward and 
backward set of pointings and readings made. These form a 
set. The circle is then shifted into a new position and another 
set observed, as already described. The average of the direct 
and reversed readings of each series is taken as a single deter- 
mination of a direction. 

Let x be the angle between the 
zero of the instrument and the direc- 
tion of the first line, A, £, C, etc., 
the angles the other lines make with 
the first, whose most probable val- 
ues are to be determined, and let 
m v m^m, ... be the reading of the 
circle when pointing to the signals in order, of which x x is the 




CALCULATION OF THE TRIANGULA TION. l8l 

most probable, and the errors of observation m x — x v Suppos- 
ing no errors existed, we should have the following equations: 

m x — x x = o ; m x — x x — A = o ; 
m* — x x — B = o ; m x — x x — C = o. 



The second series would give 



;/z 2 — x^ = o ; m* — x a — A = o ; 
7/z 2 2 — ^r 2 — .5 = o ; ?// 2 3 — x % — (7 = o ; 

and the /zth, *#« — x n = o ; #2« x — #"* — -4 ■= o ; 
tn* — x n — B — o\ in,?— x n — C = o. 

The most probable values will be those the sum of the 
squares of whose errors is a minimum. Also, the errors 
squared must be multiplied by the corresponding weights,^, 
p x ,p? . . 'pupi • • • which will give 

pfym, - *,)' +A'« - *, - A)" +A'« - *, - Bf 

A(«, - x,y +/,'(< - *, - il)" +A'« - *. - £)' 

+A°« - * 2 _ C)'; 

AK - *,Y +A'« - *. - Ay +/,'« - *. - *)' 

etc., etc. 

Differentiating with respect to x v x,, x, . . . A, B, C . . . and 
placing the differential coefficients separately equal to zero, we 
shall have 



1 82 GEODETIC OPERATIONS. 

p.m, +/i'**i' +Pi m i + Pi m i ■ ■ ■ 

= (A +Pi +Pi +/," ■ ■ ■ fr+tW+A'B+A'C ■ ■ ■ 

pj'h +/.'«.' +A ! < +AX a • • • 

= (A + A' + A 2 + A 3 • • • fr+fi'A+fi'B+XC •••;}. (A) 

A»*s+A ,; < +Pi'»i +/X • • • 

= (A +A' +A' +A ! • • • K+A'^+A^+A'C 

etc., etc.; 



pint? +A'm* +Pi m i + ■ ■ ■ 
= (A' + Pi +Pi ■ ■ ■ ) A +Pi*> +Pi*< +Pi*> ■■.; 

p>i +p>i +p>i + ■ ■ • 
= (Pi +Pi +Pi ■■■)£ +/>, +av. +/,•*. • • • ; 

/,'«.' +Pi m i +Pi m i + ■■■ 
= (pi+pi+pi • • • )c+Pi*. +/."*. +/,'*, — 



MB) 



In these equations x x — m lt x 2 — m 9 , x 3 — m 3 . . . are the 
errors of observation ; calling these x v x 2 , x % . . . they will rep- 
resent the corrections of the first, second, third . . . pointings 
from the zero-mark — usually a small quantity. 

By multiplying out the parenthesis in the second member of 
(A), and transposing all the terms from the first, we have 

O = pjc x — p l ^ l -\-p x x l — p l 1 m 1 1J r p l 2 x i — p?m*-\-p*x x — P*m* 

+ A >A+p i '£+J>;C...; 

o =P,(*,-"*i)+Pi(Xi-f>ti)+Pi(x,-mi)+pi(x,-mi) ■ • ■ 

+A >A+A'B+pX- 

Introduce into each parenthesis m x — m v except the first, 

o = p 1 {x 1 — ^ 1 )+/i I (- r i — m \ ~ m i~{~ m i)-\~Pi\ x i —m 1 i -{-m J —m 1 ) 
+Pi{*-m l '+m l -m l ) . . . J^p^A+p^+p.'C. . . ; 




CALCULATION OF THE TRIANGULA TION. 



o = Ate - W .)+A' [te - »0 - K 1 - «0] 

+A c [te-^ 1 )- = t^ 1 2 -^i)]+A 3 [te-^ 1 )-K 3 -^0] 

+A^+A 2 ^+A 3 ^... 



For ^ — m x substitute x v and for m x l — m x write ;;//; re- 
membering that «/, which is to take the place of;;// — m tl does 
not mean the tth. reading on the st\\ arc, as recorded, but the 
recorded reading minus the reading of the zero on that arc. 
i This will reduce the last equation to 



o =A*, +Ate -AX 1 + Ate -A 2 <+Ate ~AV 



= (A + A 1 + A 2 + A 3 te+A^ +A 2 ^+A 8 ^. - 
In the same manner the other equations (A) reduce to 

= (A +A' + A +A' • • .)*.+t;A+f,*B+p,'C . . 

a'< +a< +a 3 < . ■ . 
= (a + a 1 + a* + a 3 • • • )*,+p;a+p;b+px . . 



(C) 



Likewise, equations (B) reduce to 

AX'+AX'+A'**,'- •• 

= (A" + A* +A' • • • M + A'*, +A**, +A 1 *. • 

= (A+A +A -..")* +/,'*, +AX +A 3 *. • 

AX 3 +/X + AX 3 • • • 

= (A 3 +A+A 3 • • • )C J-A'*. +A 3 * 5 +AX • 



(D) 



When the signals observed upon are numerous, the solution 
of equations (C) and {D) would be very laborious. 



184 



GEODETIC OPERATIONS. 



Captain Yollond, of the Ordnance Survey of Great Britain, 
found the method of successive approximations sufficiently 
accurate. 

Suppose x v x„ x z . . . severally equal to zero in (D), from 
which we find the first approximation : 



, _/>i'+ AX 1 +/>,'■■. 



A' = 



A'+A'+A 1 -- 



/. /X+/X+/X... . 



#' = 



A a +A a +A a .. 



" A 8 +A 8 +A 3 ... " 



Substituting these values in (C), we obtain a new value for x lt 



/.■(w.'-^HA^,'- B') + p,°(m:- C) . . 



A+A'+A' + A 3 -- 



x, 



A+A'+A' + A 3 --- 



_ _ /.'«- A') + A'«- ^) + /,'«- CQ ■ ■ ■ 
A+A+A 2 + A 3 --- 

Substituting these values in (D), we obtain the second approxi- 
mation, or 



A" = 



p x \m x x - x x ) + pj{mj - x,) + p t \m 2 \ - x % ) . . . 
A'+A'+A 1 -.. 



CALCULATION OF THE TRIANGULATION. 1 85 

_ /,'« - .*■,) + p,\m; - x,) + a'K' - »,)... . 
A'+A'+A*.'.. 

_, _ /,'(;«,' - *,) + A'K* ~ *,) + A 3 K 3 - *,) ■ ■ ■ 
A'+A+A'-.- 

The values can be further substituted in (C) and the result- 
ing values of x lf x„ x z . . . placed in (D) for the third approxi- 
mation for A, £, C . . . However, the second has been found 
sufficient in good work. 

The weights for observed directions is unity, and zero for 
any directions that could not be observed. The work can be 
materially shortened by pointing on the first object on the left, 
as the beginning of each series; and in each successive series 
the readings of the first direction should be diminished by the 
preceding direction, in this way taking as a zero the first direc- 
tion of each series. 

In the ordnance survey, the readings on the initial object 
were made the same in the different series by adding to the 
average readings of the microscopes on each signal such a 
quantity, positive or negative, as to make the initial readings 
the same. 

Considering the weights unity, 

_ m* + m* + m* ... 1 



where n represents the number of series, or A f = the arithmet- 
ical mean, say M x \ in the same way we find 

B' = M» C = M 2 ... 

Substituting these values in the expressions for x v x 2 . . . we 
have 



1 86 GEODETIC OPERATIONS. 



x x = -(ml — M x + ml - M, + ml -M,. . .), 

lb 



or, - x x = \{M X - ml+ M % - m;+ M % - ml ...)=- Ml, 



and similarly for x v ^ ... we get — Ml, — Ml . . . 
Placing these values in the second approximation, 



A" = -(ml - Ml + ml - Ml + ml -Ml . . . ) 



B" = -(ml - Ml -f- ml - Ml + ml - Ml . . . ) ; 



C" = -{ml - Ml + ml - Ml + ml - Ml . . . ). 



We have first obtained a constant reading for the initial di- 
rection, either its angular distance from an azimuth-mark, or 
by making the first direction zero. We then found the aver- 
age of each direction, giving A' == M x , B' = M^ ... or the 
arithmetical mean as the first approximation. Next we sub- 
tracted each average from each reading, giving a set of errors 
— the average of those in the same series giving Ml, Ml . . . 

Afterwards these are taken from the readings of the corre- 
sponding series, giving diminished values of each direction ; 
and the average of these diminished directions gives the second 
approximation. 

A symbolic analysis can be seen in the appended table, fol- 
lowed by an example taken from the Report of the Ordnance 
Survey, 1858, page 65: 



CALCULATION OF THE TRIANGULA TION. 



137 



Initial Object. 


A. 


B. 


C. 


Averages. 


VI 


Wl 1 


m-? 


m^ 




m 


Vl2 X 


m<? 


w 2 3 




m 


Jllz X 


*»s 9 


ot 3 3 




Average 


Mi 


m, 


M 3 






m, 1 - Mi 


mj — Mi 


W1 3 — M z 


Mt l 




ffl 2 ] — Mi 


w 2 ' 2 — M 2 


»/j 3 — M 3 


M 2 l 




ms 1 - Mi 


m 3 2 - M 2 


w 3 3 - Mz 


M s * 




mj - M x x 


w, ! - M x l 


m x z — M x l 






mj - MJ 


m<? — MJ 


w 2 3 — M 2 l 






m z l - J/3 1 


;// 3 2 - M 3 L 


w 3 3 - Ms 1 




Averages — 


A" 


B" 


C" 





No. of 
Series. 


Initial 0. 


A - n° 7'. 


B = 37° 34'- 


C = 97° 54'- 


D = 220° 3'. 


Average 
Errors. 

-f- l".IO 
+ .73 

— O .23 

— I .19 

— O .60 
+ .67 


I 
2 

3 
4 

5 
6 


4°2l'29".2I 
29 .21 
29 .21 
29 .21 
29 .21 
29 .21 


36' '.04 
35 .91 
34 .21 
32 .41 


I4".07 


47". 84 


I9". OO 
18 .18 


11 .86 

10 .71 

11 .91 




46 -OS 

48 .30 


16 .30 
14 .17 
18 .59 








Average 


29". 21 


34". 64 


12". 14 


47"-40 


i7"-25 


Errors. 
Average 


OO". OO 
.00 
.OO 
.OO 
.OO 
.OO 


+ i"-40 
+ 1 .27 

— .43 

— 2 .23 


+ i"-93 


-f o".44 


+ i".75 
+ .93 


— .28 

— 1 -43 

— .23 




- 1 -35 
+ .90 


- .95 

- 3 .08 
+ 1 -34 










28". II 

28 .48 

29 .44 

30 .40 
29 .81 
28 .54 


34". 94 
35 .18 
34 -44 
33 .60 


12". 97 


46". 74 


17". 90 
17 45 




12 .09 

11 .90 

12 .51 




46 .28 
48 .90 


17 -49 
14 -77 
17 .92 










2 9 ".I3 


34". 54 


I2".37 


47"-3i 


I7".IT 



1 88 GEODETIC OPERATIONS. 



This gives the directions as follows: 



Initial object == 4 


2I , 2 9 ,; 


.13; 


direction A" = 1 1 


7 34 


.54; 


direction B" = 37 


34 12 


•37; 


direction C = 97 


54 47 


.31; 


direction D" = 220 


3 17 


.11. 



The third approximation, obtained in the same way, gave, 
omitting degrees and minutes: initial object = 2o/ / .i2, A!" = 
34".55, B"' = I2".40, C" = 47 // -34, ^ ,,/ = i?''x&, values dif- 
fering from the above in the hundredths place only. 

The angles depending upon these directions will be involved 
in the figure-adjustment, so their corrected values should be 
written A + (1), B + (2), C+ (3) . . . in which (1), (2), (3) . . . 
are the corrections obtained in the figure-adjustment. In this 
operation the directions obtained at different stations have not 
the same weight ; however, this can be computed from the 

formula already given on page 125, where we found p = — j- 



So we find the residuals by taking the difference between 
the individual diminished measures and the average, and di- 
vide the number of readings on that direction squared by 
twice the sum of the squares of the residuals ; in the case of 

16 

To illustrate the formation of the equations of geometric 
condition let us take an example. 
The angles adjusted at stations are : 



CALCULAT10X OF THE TRIAXGULATIOX. 



189 




at T, M — ocr 00' 00" 

F = 83 30 34 .866 + (1) 

1V= 2S7 14 13 .822 + (2) 
at 31, T — 00 00 00 

W= 66 56 10 .619 + (4) 

F= 293 5; 16 .395 +(6) 
at F, W = 00 co 00 

M — 20 co 09 .436 -f- (7) 

T= 349 33 27 .528 + (11) 
at W, F = 00 00 00 

r= 13 17 05 .983 +(12) 
^f = 332 59 01 .843 + (15). 

In the triangle MTF, we are to find the angles at each ver- 
tex, as follows : 

8 3 °30'34 // .866+(i) = MTF- 

66 243 .60$— (6) — FMT, or 360 — direction 

Firom T; 
302641 .9oS+(7)-(u) = MFT; 

i8o°co / oo^379+(i)-(6)+(7)-(ii)= sum; 

180 00 00 .01 5 = 180 + spherical excess ; 

o = o". 3 64 + (1) - © + (7) - (1 1). Equation (I) 

To find MFT, we subtract the direction of T from W, from 
360 ; this gives angle WFT\ to this add the direction of M 
from W, or the angle WFM. 

In the triangle TMW, 

72°45 / 46".i78-(2) = MTW 

66 5610 .6i 9 +(4) = WMT 

40 18 4 .i40+(i2)— (15) —MWT\ 

i8o°oo'oo / '.937-(2)+(4)+(i2)-(i5)= sum ; 

180 0000 .011 = i8o°-f-spherical excess ; 

Equation (II) 



O = + 0". 9 26 _(2)+(4)+(l2)-(l 5). 



190 GEODETIC OPERATIONS. 

In the triangle WTF t 

I3°i7'05".983+(i2) = TWF 

156 16 21 .o44+(i)-(2) = WTF 

102632 .472— (11) — TFW 

i79°59 , 59 // 499+(0-( 2 )-( II )+( 12 ) = sum; 

180 0000 .009 = 1 8o°-)-spherical excess; 

o = - o".5io + (i)-(2)-(i i)+(i2). Equation (III) 

In the quadrilateral TMFW, the side equation is 

__ sin TMW. sin FWT . sin TFM 
1 ~ sin MWT. sin TFW . sin FMT ; 

sin TMW = 9.9638207,6 + 8.965(4) (8.965 = tab. dif.) ; 

sin FWT — 9.3613403,1+ 89.174(12); 

sin TFM = 9.7047600,1 -f 35. 824^7) - (11)] ; 

29.0299210,8. 

sin^/^r= 9.8107734,2+ 24.826K12)— (15)]; 
sin TFW = 9.2582687,7— 114.245(11); 
sin FMT = 9.9608833,6 - 9-354(6); 

29.0299255,5. 

o = - 44-7 + 8.965(4) + 9-354(6) + 35*824(7) + 78421(1 1) 

+ 64.348(12) + 24.826(15). Equation (IV) 

These four equations are solved for the unknowns, which 
are applied to the given directions with their proper signs, or 
to the angles directly, as just deduced. 



CALCULATION OF THE TRIANGULATION. 



19] 



In an extended triangulation, the position of every point is 
influenced to a certain extent by the directions at the adjacent 
signals ; consequently, it is advisable to include in the equations 
of condition as many directions as possible. The influence of 
these directions upon an initial point diminishes with the dis- 
tance, and finally becomes inappreciable, so that the triangula- 
tion can be divided into segments, each containing a conven- 
ient number of conditional equations. The corrections of the 
first are computed, and, as far as they go, these corrected val- 
ues are substituted in the equations of condition in the second 
figure, and the sum of the squares of the remaining errors, each 
multiplied by its corresponding weight, made a minimum. 

The equations of condition (I), (II), (III), (IV) . . . may be 
written 



o = a + a x x x -f- ajc^ . . . 
o = b + b x x x + b,x, . . . 
o = c -f- c x x x + c^x\ . . . 



(E) 



^ Pv A • • • be the weights, corresponding to the corrections 
x x , x 2 . . . , the requirement that the sum of the squares of the 
errors be a minimum is 



p x x? -\-p 2 x^ + P% x * . . . = a minimum. 
Differentiating (E) and (F), we have 



(F) 



o = a x dx x -f- a^dx^ -\- a 3 dx 3 . . . 
o = b x dx x + b^dx^ -f- b 3 dx s . . . 
o = c x dx x + c 2 dx t -f- c 3 dx % . . . 



=Pi x idx x +p i x i dx i ^p z x % dx % 



I92 GEODETIC OPERATIONS. 

Solving these equations as explained on page 160, we have 

p,x, = a J x + bJt + cJ t ...;> . . . (G) 
p,x z = aj x + b % I % + cj 3 ) 

Substituting the values of x xi x„ x z ... as found in these equa- 
tions in (E), we have 



o = a + J(«/,+V. W. • • • ) + £(«/. W.W. • • • ). 



In the same way, remembering that (a?) is the sum of the 
squares of quantities like a, as a x -\- a*-\- a s \ . . and (ad) = tf,^ 
+ a A + "A • • • , 

-»+ $■.+$« +$<,•■■ . 

°=«+(f>,+( 7 )'•+ (?)'■■ •■ • 



(H) 



/„ / 2 , 7 3 . . . , being auxiliary multipliers, have their values ob- 
tained from (H) and substituted in (G), giving the numerical 
values of x v x» x s . . . 

Instead of using I lt I t . . . the Roman numerals I, II, III . . . 
will be found more convenient, especially when the conditional 
equations are so numbered. The normal equations can be 
more readily formed. 

To illustrate, suppose we have the following equations of 
condition : 



(I) 



CALCULATION OF THE TRIANGULA TION. 1 93 

I, O = - I. 4 4 2-(2) + (5) - (7) + (3) ; 
II, 0= - 2.7737-(2) + (4) ; 
III,o=-0. 9 595-(8) + (ii); 
IV,o = '-i.2i57-(3) + (4); 
V, o = - o. 9 204-(3) + (5) - (7) + (10) ; 
VI,o=-o.8 4 2 4 -(i) + (4); 
VII, o = - o. 3 2oi-(i) + (5) - (7) + (9) ; 
VIII, o -+0.999 -(1) + (3); 



X,o=- 4 -0567-(6) + (9); 
XXV, 0= + 3-298o-o.oooi5(2)+i5.57i 9 (4)-i5.57i( 5 ) 

-6.188(7); 
etc., etc. 



(1), (2), (3) . . . represent the corrections to directions of the 
same number ; then we multiply the terms involving (1), (2), 
by the reciprocals of their weights, giving 

(1) = — 0.0800VI — 0.0800VII — 0.0800VIII 

((1) occurs in VI, VII, and VIII, and 0.800 is the reciprocal of 

its weight) ; 

(2) =— 0.2060I — 0.2060II — O.OOO0309XXV ; 
(3)=— 0.1 580IV — 0.1580V + 0.1 580VIII; 

(4) -+0.3380II+0.3380IV+0.3380VI+5.263302XXV; \- (K) 

(5) =+o.226oI +0.2260V + 0.2260VII — 3.51922XXV ; 

etc., etc. 

These values of (1), (2), (3) . . . are substituted in the equa- 
tions of condition (I), giving numerical values for I, II, III ... ; 
13 



194 GEODETIC OPERATIONS. 

then these values substituted in equations (K) give the val- 
ues of the corrections (i), (2), (3) . . . , which, when applied to 
the directions, will give their most probable values, satisfying 
the geometric conditions. 

For the various methods of adjustments, see : 

Jordan, Handbuch der Vermessungskunde, vol. i., pp. 339— 

346. 

Bessel, Gradmessung in Ostpreussen, pp. 52-205. 

Clarke, Geodesy, pp. 216-243. 

Wright, Treatise on the Adjustments of Observations, pp. 
250-348. 

Ordnance Survey, Account of Principal Triangulation, pp. 
354-416. 

C. and G. Survey Report for 1854, pp. 63-95. 

Die Konigliche Preussische Landes-Triangulation, I., II., 
and III. Theile. 

When a number of normal equations are to be solved, it is 
found, by some, desirable to eliminate by means of logarithms ; 
but, as logarithms are never exact, there will always remain 
small residuals when the corrections are applied. Direct elimi- 
nation is preferable, unless the coefficients are large ; then the 
logarithmic plan is somewhat shorter. We will illustrate with 
an algebraic equation : 

lu -f- x -f- 2y — £ — 22 = 0;. . . . (1) 

4* - 7+ 3* - 35 = o;. . . . (2) 

4* + 3* — 27 -19 = 0;. . . . (3) 

2u -f- 4y + 2z — 46 = o (4) 

If the first equation were multiplied by |-, the coefficient of 
u would be the same as in (3), and upon subtraction the u's 
would disappear. To multiply by -§- is simply adding log 4 — 
log 3 to the logarithms of the coefficients of (1), omitting 32/ ; 
we write, then, the logs of these coefficients : 



CALCULATION OF THE TRIANGULATION. 



195 



X. 


y> 


z. 


22. 




Log of coef., 0.0000 


0.3010 


no. 0000 


n 1. 3424 


= 0; 


log 4 —log 3, 0.1248 


0.1248 


0.1248 


0.1248 




add 0.1248 


0.4258 


#. 1 248 


#1.4672 




nat. numbers, 1.333 


2.666 


- 1-333 


- 29.33 




coef. of (2), 3 


— 2 


+ 


- 19 




subtract — 1.667 


+ 4.666 


- 1.333 


- 10.33- • 


(5) 



Take a factor that will make the coefficient of u in another 
equation equal to its coefficient in one of the other equations, 
multiply (4) by 2, or add to the logs of the coefficient in (4), 
the log of 2 = 0.3010. 







X. 


y- 


z. 


46. 


Logs of coef. of 


(4), 


. . . . 


0.6020 


0.3010 


#1.6627 


log 2, 




. . . . 


.03010 


.03010 


.03010 


add 






0.9030 


0.6020 


#1.9637 


nat. numbers, 




. . . . 


8 


4 


-92 


coef. of (3), 




3 


— 2 




- 19 


subtract 




-3 


10 


4 


-73- • 



(6) 



Continue to eliminate the same quantity from all the remain- 
ing equations until one equation remains with one unknown 
quantity. 

The only advantage that this method suggests is, that only 
one quantity is used as a multiplier to make the coefficients 
identical ; that factor is usually a fraction, whose log is simply 
the difference between the logs of the numerator and denom- 
inator. 

Mr. Doolittle, of the Coast Survey, has developed another 
method of elimination, which can be found in the Report for 
1878, page 115. 



196 



GEODETIC OPERATIONS. 
REDUCTION TO CENTRE OF STATION. 



With the directions adjusted it is necessary, when an eccen- 
tric position has been occupied, to reduce the corrected ob- 
served directions to their equivalents at the centre, before 
computing the distances and co-ordinates. 




.-^ c 



Fig. 25. 

= observed angle ; 
x = desired angle. 

Angle from signal to A = a, to B = b. 

Angle m = A + x = 6 + B. 

x=6-\-B-A. 

r sin (6 -\- ri) 



sin B : sin {6 -f- n) :: r : b, sin B = 

sin A = 



b 
r sin n 



CALCULATION OF THE TRIANGULA TION. 



I 9 7 



As B and A are always very small, they may be regarded as 

n . ,, , . . „ _ r sin (Q-\-n) . rsinn 

equal to B sin V , and ^4 sin 1 ,2? = — 7—: ?7i A — — ■ — r,\ 

^ ' b sin 1 «sin 1 ' 



hence, 
Also, 



x = + 



r sin (0 -|- n) r sin « 



^ sin 1 



a sin 1 



angle between B and C = AOC 



r . sin (n-\-0-\-6 ; ) r sin ^ . 



c . sin 1' 



# sin 1 



From the above equations it will be seen that all angles that 
are read from the same initial point have for their corrections 
the same last term ; so this term can be 
computed for each initial direction and 
applied to the various angles. In both 
terms of the corrections there are two con- 
stants for each station, rand sin 1".; so the 
work can be facilitated by tabulating their 
values. The signs of the terms will de- 
pend upon the sign of the sine function. 

It will assist in the computation to take 
the angle between the signal and the first 
point to the right and continue in that 
direction. 

Signal 23 feet from instrument. Angle 
between 77 and C.T — 71 . 




C.T 



Fig. 26. 



Log dist. H to C.T in M. = 47534757 ; 
log dist. H to B.K in M. — 4.6503172 ; 
log dist. H to H.K in M. = 4.8385482 ; 
log dist. H to C in M. = 4.6145537. 



198 



GEODETIC OPERATIONS. 





Sig. toC2' + 
C.TX.O B.K. 


Sig. toC. T-\- 
C.TtoH.K. 


Sig. to C.T + 
C. T to C. 


Sig. to B.K + 
B.K to H.K. 


Sig.to.S.^-i- 
B.K to C. 


Direction 

Log sin 

Log r{M). . . . 
Co. log dist. . 
Co. log sin 1" 

Cor 


107° 24' 17" 
9.9796466 
O.8457389 
5.3496828 

5.3I4425I 

I.4S94934 

30". 86 


120° 53' 55" 
9.9335264 
O.8457389 
5.1614518 
5.3I4425I 
I- 2551442 

17". 99 


1 66° 06' 57" 
9.3S01384 
O.8457389 
5.3854463 
5.3144251 
O.9257487 
8". 42 


120° 53' 59" 

9.9335214 
O.8457389 
5.1614518 
5.3144251 
I- 2551372 
17". 99 
• 


1 66° 06' 55" 
9-3801555 
0.8457389 
5.3854463 
5.3144251 
0.9257658 
8". 43 





Signal to C. T. 


Signal to B.K. 


Direction 

Log sin 


71 — OO — OO 
9.9756701 

O.84573S9 

5.2465243 
5. 3 14425 I 

I.3823584 
24". 12 


107 — 24 — 17 
9.9796466 

O.8457389 
5.3496828 
5-3I4425I 
I.4894934 
30". 87 


Log ?{M) 

Co. log dist 

Co. log sin 1" 

Cor 



CORRECTED ANGLES. 



7"+3o' 



24". 1 2 = 36° 24/ 23 /r 74 
24".i2=49 53' 4 8".87 



C.T.toB.K^ 36 24' 

Cr.to^.ir=49 53 / 55 // +i7 // .99- 
C.Tto C. = 95°o6 / 5/'+ 8 // .42-24 // .i2= 9 5°o6 / 4i // .3o; 
B.KAoH.K= if 2 9 ' 4 2 ,/ +i7 // . 9 9-3o // .87=i3 2 9 r 29 // .i2 ; 
B.K to C. = 58 42 ; 38 7/ + 8 ,, .42-30 ,/ .87=58 °42 / 1$".$$- 



The distances used above were obtained from the observed 
values of the angles, and are, therefore, only approximate. In 
the case of refined work, it will be necessary to use these cor- 
rected values and again compute the distances ; then, with the 
correct distances, recompute the reduction to centre. 

With the most probable value for all the directions, the 
angles of all the triangles can be found by taking a given di- 
rection from 360 , or by adding or subtracting two or more 
directions. 



CALCULATION OF THE TRIANGULATION. 1 99 

Then with a base, measured, or previously computed, each 
side can be found by the trigonometric formulae a — 

b sin \A — -) b. sin \C — -) 

, and c = ■ -. ~ , in which e is the 



sin \B J sin \B 

computed spherical excess, as obtained from using approxi- 
mate lengths and angles. 

If there are more than one base in the triangulation-net, the 
most satisfactory method is to compute each base from all the 
others, and take the mean of the logarithmic values so found; 
or, if the entire scheme is involved in a single figure, the abso- 
lute term in the side equation can be made equal to the ratio 

B x 

of the two bases, -5-, instead of unity. 

We can also find the length of any line as influenced by two 
or more bases. Let B iy B„ B 3 . . . be the bases, x the most 
probable value of any side in the triangulation, and r v r a , r 3 
. . . be the ratio of each side respectively to x\ the errors 
then will be 



(r x x - B x ), (r 2 x - BJ, (r s x - B s ) . . . 
Now, if A' A' A • • • be the weights of the bases, then 

A( r i x — £i)' 2 -\-A( r z* — ^ 2 ) 2 + A(V — -^a) 2 . . . = a minimum. 

Placing the differential coefficient with respect to x = o, we 
find 






200 



GEODETIC OPERATIONS. 



From this we can find the most probable length of one base 
from all the others. To do this we suppose x to be one of 
the bases, say B v then r x = I, 



the correction will be x — £ x , or 

PJB, +Ar,B, +Ar,B 3 . . . 



£,= 



B, 



A +A'.' +Ar,' 
A3, +ArA+Ar s B, - p,B, -fs.'B, -p z r?B x 

A +A',' +Ar,° ■ ■ ■ 

PSJJB, - rJB,) +fiS,(B, - r,B,) . . . 

A+Ar,*+Ar;... 



The adjustments so far considered affect the geometric con- 
ditions, and in their operations may, by changing the direc- 
tions of the lines, change the azimuth, making a greater or less 
difference between the observed and computed azimuths. In 
refined geodetic work, the azimuth is observed at least twice 
in each figure, and sometimes twice in each quadrilateral. 




Fig. 27. 



Using Wright's figure and notation, we take PQ and TU as 



CALCULATION OF THE TRIANGULATIOJST. 201 

two lines whose azimuths have been observed with the simplest 
and most approved connections. PQ, as a known line, enables 
one to compute PR, and from PR we can go direct to SR, 
thence to ST; so these lines are called sides of continuation. 
Let A x , A 2 , A % ... be the angles opposite the sides of con- 
tinuation ; 
B x , B„ B % . . . the angles opposite the sides taken as 

bases ; 
C v C 2 , C 3 . . . the angles opposite those sides not used ; 
Z lf Z 2 , the measured azimuths of PQ and TU, 

supposed to be correct, and therefore 
subject to no change ; 
Z' the computed azimuth of TU, reckoning from the 
south around by the west. C v C„ C s , C v are the only angles 
that enter into this computation ; and the excess, E, of the 
observed over the computed azimuth gives 



-(Q + (Q-(Q + (C,) = £, . . Eq.(L) 



in which (Q), (C^) . . . represent the corrections to cT,, £7 2 . . . 
Now, since the triangles have had their angles adjusted to the 
conditions imposed upon them, their total corrections must be 
zero. 

(A,) + (B 1 ) + (Q = o;) 

{AJ + {B,) + {C,) = o;} . . . . (M) 
(A,) + (£,) + (Q = o.) 



Also, the sum of the errors squared 

{A i y-\-{B 1 ) 2 + (Q a . . . = a minimum. 



202 GEODETIC OPERATIONS. 

The solution of these equations would give 

A,= IE, A t =-iE...; 
B, = %E, B,= — IE . . . ; 
C X =-±E, C, = + $E.... 

If there were n intervening triangles, we would find 



A,= --E, A,= ---E; 
B >= Tn E ' B > = -Tn E > 



From which the following rule is deduced : 

" Divide the excess of the observed over the computed azi- 
muth by the number of triangles, and apply one half of this 
quantity to each of the angles adjacent to the unused side, and 
the total quantity with its sign changed to the third angle. In 
each following triangle the signs are reversed." 

The discrepancies between the observed and computed lati- 
tudes and longitudes are very slight, and can be adjusted arbi- 
trarily. 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 203 



CHAPTER VII. 

FORMULAE FOR THE COMPUTATION OF GEODETIC LATITUDES, 
LONGITUDES AND AZIMUTHS. 

WHEN we know the geographical position of a point, and 
the distance and direction to another, the co-ordinates of the 
second can be computed from the data just named, by using 
formulae for the difference in the latitudes, longitudes and azi- 
muths. 




In the above meridian section, let A be a point whose 
latitude is L. By definition it is equal to the angle ANE, 
formed by the normal AN and the equatorial radius EC. 

2 72 

AG = N, e* = j — , in which a is the semi-major axis, 

and b the semi-minor. 



204 GEODETIC OPERATIONS. 

The subnormal in an ellipse MN = CM ,— v 



AM = NM. tan L = CM. - a . tan Z, 

a; 



squaring, AM* = O? 2 . - 4 tan 2 Z. (i) 



The equation of an ellipse gives 



AM* = - 2 (# 2 - CM*\ 



V 



therefore CM 9 . - 4 tan 2 Z = - 2 <> 2 - £AP), 



CM*X tan 2 L = a*- CM* ; 



clearing of fractions and transposing, 

CM\b 2 tan 2 L + a 2 ) = a 4 ; 



hence CM* - 



£ 2 tan 2 Z+tf 2 ' 



sin L _ 
substituting =-=. for tan Z, 

s cos 2 Z ' 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS, 205 
c£ cos 2 L c£ cos 2 L 



CM'- 



CM = 



b 2 sin 2 L + a- cos 2 L ~ b\i - cos 2 L) + a 2 cos 2 L 

a* cos 2 L 
{a 2 - b 2 ) cos 2 L + b 2 ' 

a 2 cos L 



V(a 2 -b 2 ) cos 2 L+b 2 
From definition #V = a 2 — b 2 , 

a 2 cos L a 2 cos Z 



CM = 



VaV cos 2 Z + b 2 VaY(i - sin 2 L)+a*-a*S 

a 2 cos L a 2 cosL acosL 



Va 2 - a 2 e 2 sin 2 L "Vi-e 2 sin a L Vi-e 2 sin 2 Z' 
which is the radius of a meridian. 

AO AO a 

In the triangle A GO, AG = —. — tt^ = / == — ===== 

* sm AGO cosL Vi-e 2 sm 2 L 

= N, or the normal produced to the minor axis. 

The ordinate AM can be found from the equation of the 

ellipse, a 2 . AM 2 + b 2 . CM 2 = a 2 b 2 , 

a *V - b 2 CM 2 a 2 b 2 b 2 a 2 cos 2 L 



AM 2 = 



a 2 ' a 2 a 2 (i — e 2 sin 2 JO) 



_p_ b 2 cos*L = _ b% b\i-sm 2 L) 
i — e 2 sin 2 L I — e 2 sin 2 L 

_ p - b 2 e 2 sin 2 L-b 2 -\- b 2 sin 2 L 
i — e 2 sin 2 L 



206 



GEODETIC OPERATIONS. 



b* sin 2 Z (i - ?) _ a 9 sin 2 Z(i - ^) a # 
I — ^ 2 sin 2 Z, I — e* sin 2 Z ' 



therefore 



AM. 



a{\ — e*) sin L 
Vi — e* sin 2 Z' 



To find the normal ^iV, we take the triangle AMN, in 
which 



^iV 



*(i-* 2 ) 



AM 



sin Z ' tfi—S sin 2 Z,' 



The radius of curvature, R = 






In the general equation, ~ = - -y, [^j = -j 

substituting these values, 



/? 



- L 1 + Sy. 



yj _ a'f 



J? 

a*/ 



a y + jy]* _ py + fl 4 **) 1 



^ 4 L *y "J 



^ 4 



In this expression we place for x the value we found for CM, 
and for/ that of AM; this gives 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 20/ 

r y(! _ e y sin 2 L a\\ - ej - a\i - ej sin 2 L 
L I — e 1 sin 2 L I — e % sin 2 L 



R = 



a 4 b 4 



J a%i 



Li-^ 2 sin 2 Zj a\i - ej 



R = 



a K V ~~ a\\ - ej ' (i - e* sin 2 Lf 



(i-*«sin a Z)*' 



The terminal points a, b, c, d and e of the radii of curva- 
ture form an evolute ; at the equator, 



L = o° y sinZ = o, R = a(i — ^) = -. 

u 



At the pole, 

tf(i-, 2 ) 



L = go°, R 



(i - ^ 2 )t — (i - e'J ~ b 



The above formulae are in terms of geographic latitude; 
the geocentric latitude is equal to the angle formed at the cen- 
tre by the equator and radius. In the figure it is the anp-lf* 
ACM. Calling it d, we have 



_ AM _ a(i — e 1 ) sin L a cosL 

tan MC~ (i - e" sin 2 Lf ^ (i - S sin 2 Ly 

(i-V)sinZ , T b* 

— ~ = (i — e>) tan L-- tan L. 

cos L J a 



208 



GEODETIC OPERATIONS. 



It is always less than the geographic latitude, the difference 
being greatest at those places where a 2 — b 2 is the greatest, or 

at latitude 45°, N. or S., where the 
difference is about Ii' 30". 

In the adjoining figure, Pis the 
pole, E the plane of the equator, 
A and B two points on the earth's 
surface whose latitudes are L and 
L\ co-latitudes A and A', and the 
geodesic line AB is /. An and 
Bn\ the normals, are N and N', 
and R and R! the radii of curva- 
ture. 

The azimuth is estimated from 
the south around by the west; the 
angle PAB — 180 - Z, will be 
designated x, and the angle be- 
tween the two meridians AP and 
BP is the difference of longitude, 
dM. 

In the spherical triangle APB, 




Fig. 29. 



cos V = cos A cos l-\- sin A sin I cosx. 

In relation to A and A', / is very small, so that a series involv- 
ing / will converge, so we write A' = /(A + 0- 
Developing this by Taylor's formula, we have 



dX d 2 X d*\ 



dl 



2.dl- 



2.$.dP 



• ■ (1) 



In order to find these differential coefficients, some relation 



must be established between A, \-\-d\ and dl. 



Taking these 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 20O, 

as sides of a differential spherical triangle having the angle 
between the first and third sides, we have 



cos (A -|- dX) = cos dl cos A. -J- sin Asin^/cos x. 

Obtaining the differential coefficients of A with respect to /, 
we substitute them in (1) ; this is accomplished by expanding 
the last equation so as to have 

cos A cos dX — sin X. sin dX = cos dl cos X 

-f- sin A. sin dl cos x ; 
cos dX = I, 

and sin dX = dX. This reduces the equation to 

cos A — sin X dX = cos Xdl-\- sin X dl cos x. 

Cos X may be assumed equal to cos X . dl, and therefore 

dX 

they eliminate each other, leaving —77- = — cos^r. 



d 2 X d*X . dx , dx 

—rr = smx. dx —rr^ = sin x -77 ; but -77 = — sin x . cot A 
dl dl dl dl 

d*X d % X 

-777 = sin 2 x cot A, also -ttt = sin 2 x cos ^(1 -f- 3 cot A). 



Substituting L and Z for A ; and A, and remembering that cot 
X = tan L, we obtain from (1), 

L'—L = —I cos x — \P sin 2 x tan L 

-f- -i-/ 3 sin 5 ;tr cos ;r(i + 3 tan 2 Z), 
14 



210 GEODETIC OPERATIONS. 

x = i8o° — Z, cos x = — cos Z y sin x — sinZ, 
L -L=-dL, 



L - L = IcosZ + iPsm'ZtanL 

- %P sin 2 Zcos Z(\ + 3 tan 2 Z); 

or, — dL = I cos Z -f- ^/ 2 sin 2 .2" tan Z 

- -J-/ 3 sin 2 Z cos Z(i + 3 tan 2 Z). 

The value of / has been considered as expressed in arc, 
while in computation it will be given in linear measure. There- 

IC 

fore / = -jT T , where K \s the length of the line, and N the ra- 
dius of the imaginary sphere on which Z is a point 



, r KcosZ , iT 2 sin 2 ZtanZ K'sitfZcosZ, , N 

_^ = ___ + _ _ (i +3 tan'Z). 



This needs a further transformation, to refer the formula to 

an ideal sphere whose radius is the radius of curvature of the 

middle meridian. This, however, cannot be known until L' is 

computed ; however, we can start with the value of R for the 

initial latitude, and apply a correction. The reduction is made 

N 
by multiplying by the ratio of -r,-; we also divide by arc \" to 

convert the arc dL into a linear multiple of \" . This gives 



~ MW^TF sin ' Z cos Z( l + 3 tan ° z >- & 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 211 

Denoting the radius of curvature of the mean meridian by 
R„ n dL must be increased by ~ — - . dL; 

■K-m 



(I - e 2 sin 2 Z)i (i - e z sin 2 L m f 

_ (i - ^ sin' Z w )* - (i -^ 2 sin 2 Z> 3 
- a \ l e ) (i _ e * S i n « Z)i(i - e* sin 2 Z w )* ' 



Expanding by binomial formula, 



(i — e* sin 2 L m )* = I — %? sin 3 L M + \e" sin 4 L m . . . 
{i-e 2 sin 2 Z)* = i - f ? sin 2 Z + i* 4 sin 4 L m . . . ; 



subtracting = J-^ 2 sin 2 Z — f * a sin 2 L m . . . , 



omitting higher powers of e; or, J-^ 8 (sin 2 Z — sin 2 Z w ), 
This can be reduced as follows: 



sin (Z — L m ) = sin Z cos L m — cos Z sin L m ; 

sin (Z -f- Lm) = sin Z cos L m -\- cos Z sin L m ; 

sin (Z — L m ) sin (Z-|-Z w ) = sin 2 Z cos 2 Z ;;z — cos 2 Z sin 2 Z w 

= sin 2 Z(i— sin 2 Z w )— (i— sin 2 Z) sin 2 Z ;;z 

= sin 2 Z — sin 2 Z w . 

L m is the mean latitude between Z and Z -f- dL, 

Lm = i(L + L + dL), 
sin (Z — L m ) sin (Z -j- Z,„) = sin (Z — Z — \dL) sin (2Z -f- \dL) 

= dL sin Z . cos Z, nearly. 



212 GEODETIC OPERATIONS. 

^ t R — Rm 

Then, — ^ 

_ 3 2 ail - e*)dL . sin L . cos L (i — e* sin* L,„)5 
" ** (I -* 2 sin 2 Z)§ . (i -e 2 sin 2 Z>) §>< a(i - 7) 
_ g ^dL . sin L . cos Z 

~ I* (i -Vsin'Z)* * 

As this is a small quantity, it can be converted into a linear 
function by dividing it by arc i", giving 

R-Rm JT , , (dL)* sin L . cos L 
-dL = %e 



R m 2 (i - * 2 sin 2 Z)i arc i" # 

Introducing this into (2), we have, after placing 



7? - * r _ tan L _ 1 -f 3 tan 2 L 

7?.arci"' *"~ zR.N.zxol'" 6N 2 ' 

K _ _ f ^ 2 sin Z, cos Z 

77 . cos 2T, /y = 



-~~ .# . arc 1" ' ' (1 - f sin 2 Z)J arc i /n 

-dL = Z"cos Z.B + K* sin 2 Z. C- /^ 2 sin 2 Z. £+£> . (aTZ)\ 

The last term was devised by Professor Hilgard, in 1846. 

The factors B, C, D, JS, are given in the last pages com- 
puted for Clarke's (1866) Spheroid. 

When the line is not more than fifteen miles, the third term 
can be omitted, and If put for (dL)*, giving as an abbreviated 
formula 

- dL = Kcos Z. B + K* sin 2 Z.C + h'D. 

Francoeur has given a purely trigonometric method for de- 
riving the formula just obtained. 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 21 3 

Using the same figure, we write PA = 90 — Z, PB = 90 
-L. 

In the spherical triangle Z^j5, we know /M, y4i? = /, and 
the angle PAB = 180 — Z. 

cos PB = cos PA cos /+ sin A4 sin / cos PAB ; 
cos (90 — 27) = cos (90 — Z) cos / 

-f- sin (90 — Z) sin /cos (180 — Z) ; 
sin Z' = sin L cos / — cos L sin / cos Z. 

Subtracting both sides from sin Z, 

sin Z — sin L 1 = sin Z — sin Z cos / -|- cos Z sin / cos Z 
= sinZ(i — cos /) + cos Z sin /cosZ 
= 2 sin Z sin 2 \l -\- cos Z sin / cos Z. 

sin Z — sin L = 2 sin J(Z — Z') cos J(Z -f- Z') ; 

suppose L — L — d, then L -\- L' = 2Z— (Z— Z') =2L—d, 

and J(Z - Z') = Id, \{2 L-d) = L- id, 

therefore sin Z — sin L = 2 sin-J<^cos(Z — -J^/) 

= 2 sin -§^/(cos Z cos id-\-sm L sin -|dT); 

that is, 2 sin -J^ cos \d cos Z + 2 sin 2 J^/ sin Z 

— 2 sin Z sin 2 \l -\- cos Z sin / cos Z. 

Dividing by 2 cos Z cos 7 id, we obtain 

sin -|^ sin 2 id sin L sin Z sin 2 -J/ cos Z sin /cos Z" 
cos 1^/ *~ cos 2 id cos Z ~~ cos Z cos 2 -J^ 2 cos Z cos 2 ^ ' 



214 GEODETIC OPERATIONS. 

. _ , _., , _ tan Z sin 2 ^7 , sin/ cos Z 
tan J</+ tan Jrftan L = —^jg- + -J^fF 



tan id(i -f- tan ^/tan Z) = ^-y-%(tan Z sin 3 \l-{-\ sin / cos Z). 

COS 2^* 

Placing Zf for the last parenthesis, we may write 

TT 

tan \d{\ -f- tan \d tan Z) = ■ — tj~j — H{\-\- tan 2 \d) ; 

COS 2^ 

tan id + tan 2 £aT tan Z = i/ -f H tan 2 £aT ; 

tanJ^+(tanZ-i7)tan 2 i^=iZ. . . . (i) 

In the expression for H, I is small ; so we can write for sin / 

P P 

its serial value, sin /=■/ — ^-, also sin 2 \l = — , which will 

give 

ZT = J/ cos Z + J/ 2 tan Z - ^/ 8 cos Z. 

We must now solve (i) for \d\ for short we will put %d = 
c f and tan Z — H = h, so (i) reduces to 

♦ 

tan c + tan 2 c./i = If, or tan £ = — ; — ; — . (2) 

1 I + h . tan £ v ' 

Neglecting h . tan c, we have tan c = J7,3.s the value of first ap- 
proximation. Substituting this value for tan c in the second 
member of (2), 

TT 

tan c — — ; — j — f>= Zf — /*Zf 2 , by division. 

\ -\- h. H ' J 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 21 5 

Again, substituting this second approximation in (i), we have 
as the third approximation 

tan c = H - kH* + 2tfH % ; 

by continuing to the fourth, we get 

tan c = H- hH 2 + 2tfH % - S^H 4 . . . 

The development of an arc in terms of its tangent gives 

c = tan c — i tan 3 c -f- \ tan 5 c . . . 

Placing in this the value of tan c, just found, we have 

c = H-/iH> + 2h 2 H 3 - $#&' - i(H - hH* . . . ) 3 , . . 
=? H - h'H* + (2// 2 - i)H> + (i - &)hW . . . 

Resuming our notation, we have c = id, and h = tan Z 
— H\ this gives 

h = tan L — \l cos Z — J/ 2 tan Z + T y 3 cos Z ; 
\d— -J/ cos Z-f- J/ 2 tanZ — T 1 ¥ / 3 cosZ 

— (tan Z- J/cos Z — J/ 3 tan Z+ T 1 _/ 3 cosZ. . . ) 
(i/cosZ+i/ 2 tanZ - T y 3 cosZ) 2 . . . 

Multiplying this out and retaining terms of /to / 3 , 

id=iicosZ+ i/ 2 tanZ — J/ 2 tan Zcos 2 Z- -^Z 3 cosZ 

+ T V/ 3 cos 3 Z- J/ 3 tan 2 Z cos Z+ i/ 3 tan 2 Z cos 3 Z 
= i/cosZ+i/ 2 tanZ(i - cos 2 Z) - T y 3 cosZ(i - cos 2 Z) 

-i/ 3 cosZtan 2 Z(i - cos 2 Z) 
= J/ cos Z -\-\P tan Z sin 2 Z — T y 8 cos Zsin 2 Z 

— J/ 3 cos Z sin 2 Z tan 2 Z 
= i/cos Z+ J/ a tan Z sin 2 Z- T y 3 cos Zsin 2 Z(i+3 tan 2 Z) ; 



2 1 6 GEODE TIC OPERA TIONS. ' 

d — /cos Z -f- J/ 2 tan Z sin 2 Z — -J/ 3 cos Z sin 2 Z(i -f- 3 tan 2 Z). 

Remembering that d = L — L', we have here the identical 
formula given on page 210. 

There is still another form to which this can be reduced, in- 
volving more factors that can be tabulated, and at the same 
time occurring in the computation of longitude and azimuth. 

^ t , \ ' , , • „ r K K(\ -e 2 sm*L)i 

Take (2) and substitute u for ^ . ,, = : -n ■ ; 

w N sin 1" asm 1" 



multiply this by ■_-, it becomes - % — ; reducing this frac- 



N . , i-^ 2 sin 2 Z 

-pc, it becomes — 

K. 

tion and omitting terms above e 1 , 

— i-j-* 2 _/sin 2 Z = 1 +e\i — sin 2 Z) = 1 + ^ 2 cos 2 Z 
The first term becomes 

(1 -\- e* cos 2 L)u" cos Z; 
likewise the second term 

- (1 + e* cos 2 Z) (u" sin ZJ tan Z -^- ; 

substituting, it becomes 

Z'=Z-(i+/ cos 2 L)u" cos Z- i sin i^i+^cos* Z) («" sin Z)\ 

This formula gives good results for distances of twenty miles 
and under. 

The algebraic sign of the different terms depends upon the 
trigonometric functions of Z. 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 217 

Whenever the sides are more than a hundred miles long, this 
method of difference of latitude will introduce some errors. 

In that case, the method to be followed is to solve the 
spheroidal triangle formed by the two points and the pole, in- 
volving as trigonometric functions the sought co-latitude, azi- 
muth and difference of longitude. 



LONGITUDE. 

Referring to the figure on page 208, and using the same no- 
tation, we get in the triangle ABP, sin A' : sin x : : sin / : sin dM. 

If 

Supposing the radius of the sphere to be Bn' = JV } I •=. — , 
and that /and dM are proportional to their sines 



/ . -« ■•■■ •& Tn n „, K.smZ 

sin A : sin Z\\ -^ : am arc 1 , am = -^ - T - Tn 

N N cosL arc 1 



V and L' being complementary, sin A' = cosL'. 

If very accurate geodetic computations are to be made, a 
small correction must be applied, owing to the difference be- 
tween the arcs and sines of small angles. This correction can 
be taken from the table on page 274. 

The quantity dM increases towards the west. The alge- 
braic sign of the equation depends upon sin Z, which is -\- be- 
tween o° and 180 . 



// K ... « sinZ 

If we place u = -r^-. 77, dm = r~ 

r i^sini"' cosZ 



this, however, supposes that -~ = — n which is only approxi- 
mately so. 



218 GEODETIC OPERATIONS. 

AZIMUTH. 

The initial azimuth of the base or initial line being known, 
that of any line emanating from either extremity can be known 
by adding to or subtracting from the azimuth of this base the 
angle between the two lines. 

Let A B be the base, and C a point making angle m with 
ABdXA: if the azimuth of AB — Z, that of AC will equal 
Z "± in. But in order to determine the direction of a line ex- 
tending from C, as CD, the azimuth of CA must be known. 
If the earth's surface were a plane, Z would equal 180 + Z ; 
but the spheroidal shape of the earth complicates this as well 
as all other geodetic problems. 

Again referring to figure on page 208, in the triangle APB, 
by Napier's Analogies, 

tan \dM : cot \(x + x') : : cos J(A' — A) : cos i(X + A'), 

from which cot Ux + x') = C ° S ?L, J tan \dM '; 
x ' J cos \{k — A) 

but x — 180 — Z, 

therefore x + x' = 180 - Z + x' = 180 + (*' - Z) ; 

cot £[(180° + (*' - Z)] = - tan \{x' - Z), 

but ix r — Z) = dZ y also A = 90 — Z, and A' = 90 — L' , 

therefore A + V = 180 - L - L = 180 - (Z + Z') ; 

A 7 - A = 9 o° - Z' - (90 -L) = L-L' = dL; 

cos i[(i8o° -(L + £')] = sin £(Z + /■') = sin Z 

the formula then reduces to 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 2ig 

tan UZ = - tan \dM ^f'Jl . 
cos \dL 

Supposing that tan \dZ : \.a\\\dM :: dZ : dM t 
-dZ=dM 5i 



cos ^>dL 



This is not exactly correct ; the correction can readily be found 
by adding a term, say x, to the fourth term of the above pro- 
portion and solving for the value of x. It will be found that 
■Y^dM 6 cos 2 L m sinZ,„ sin 2 i" must be added to the above value 

r ,^ r~ r cos3 L m sin L m sin 2 \" 

of dZ. The factor can be tabulated as 

12 

factor F, a table of which is appended ; the expression then 

becomes 

- dZ = dM Sm f"" + dM'F; 
cos \dL ' 

Z' =Z± i3o° + ^Z. 

The algebraic sign of ^Z will depend upon dM. As the azi- 
muth is estimated by common consent from the south around 
by the west, so long as the initial azimuth is less than 180 , 
the reverse azimuth Z' = Z -\- l8o° -\-dZ\ but if more than 
i8o°,Z' = Z — i8o°+^Z. 

A table of values is given for cos \dL for lines of twenty 
miles and under. The term involving ,Fcan be omitted, and 

the value of dM deduced above substituted in its place, giving 
// • ^ 

Z' = Z ± 180 -\ ft"- sin L m . It has also been found suf- 

cos L 

ficiently accurate to omit cos \dL and write — dZ = dM sm L m . 
In accurate work the azimuth should be determined at least 
once in every figure by astronomic observation. This opera- 
tion is fully described in works on practical astronomy. 



220 



GEODETIC OPERATIONS. 



L. M. Z. FORM FOR PRIMARY TRIANGULATION. 



Mount Blue to Mount Pleasant 

Mount Pleasant and Ragged (R. is to the left of M. P.). . 
Mount Blue to Ragged (360 - (85 — 35' etc. — 26« - 19' 



Ragged to Mount Blue (Z A- dZ — 180 ) . 






/ 


26 


19 


«s 


35 


300 


44 


+ 


50 


301 


34 


121 


34 



27.01 

25.67 

i-34 
3-7i 
5-°5 
5-05 



' 


11 


43 
30 
12 


40.121 
S5-978 
44-143 



Mount Blue 

1 10740 . 6./kf , log 5.0443070. 
Ragged 








/ 


M 


70 


20 


dM — 


1 


11 


M' 


69 


08 



11.921 

27.659 



K 

cos Z 

B 



1st term 
2d term 

dL 

3d and 
4th 

-dL 



5 0443070 
9.7084622 
8.5104895 



3.2632587 



1833.406 
22.754 



1856.160 

- .182 

1855-978 

44—28—12.13 



sin*Z 
C 



3d term 
4th term 

K 

sin Z 

A' 
cos L' 
ar. co. 



dM 



10.08861 
9.86854 
1. 39991 



1.35706 



0.085 
0.267 

5.0443070 

9 934272i n 

8.5090158 
0.1446254 



99 
3.6322302 
- 4287.757 



D 



arg. 
K 

dM 
cos 



6.5372 

2-3933 



+ 3i7 
+ 99 



K* sin 2 
E 



(dM)* 
F 



dM 

sin L m 

cos \dL 

ar. co. 



— dZ 
2d term 



3 
3003 



2632 
957i 
2069 



8 9 6 n 
840 



736 
6322302 

8454305 
0000040 



4776647° 

76 

05 3°°3-7 I 



Notes upon the Computation. — The angle Mount Pleasant 
and Ragged is recorded minus, since the second point is to the 
right of the first — contrary to the graduation of the instrument. 
180 is subtracted, since the general direction is east. In the 
sixth column, — 218 and 317 correspond to the correction due 
to the supposition that the arc and sine are equal ; the value, 
99, is added to dM ; dM is negative, since sin Z is minus. In 
the azimuth-computation, the second term, .05, is the antiloga- 
rithm of 8.736 ; this is negative, therefore .05 is subtracted. 

The data here used were taken from the Coast Survey rec- 
ords. There the Z, M and Z were computed, using Bessel's 
constants: the results are, Z B = 05"55, Z c = 5".05, L B 
= 43"955, L c = 44' / .H3, M* = 43 // -578, M? = ^'.262. 

In using the abbreviated formula, the third and fourth terms 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 221 

would be omitted in latitude, but in their place should be in- 
serted hW. 

In longitude, the correction for the ratio of sine to arc is not 
inserted. Also, for azimuth-computation cos \dL, and {dM) 5 F 
are insignificant, and consequently left out. The terms that 
are disregarded could not affect the result beyond the tenth 
of a second, in lines less than a hundred miles in length. A 
very convenient form in use in the U. S. Geological Survey is 
appended, employing the abbreviated formula already given : 



Names of j p os ition. 
Stations. 


Observed 
Angles. 


Correc- 
tion by 
L.S. 


Correc- 
tions 
arbi- 
trary. 


Spheri- 
cal 
Angles. 


Spheri- 
cal 
Excess. 


Final Plane 
Angles. 


Big Knob 

Holston .. 

High Knob.. . 


Sought, 

Right, H 

Left, H 


o / // 

144 17 55.62 
13 29 28.86 

22 12 41.69 




11 
-•4 

— .1 

-.18 


55.22 
28.76 

4i-5i 


-1.83 
-1.83 

-1.83 


• 1 11 
144 17 53-39 

13 29 26.93 
22 12 39.68 



Comput- Logarithms 
_ in S 'of their Sines. 
Letter. | 


Calculation of the Sides. 


Sides in 
Yards. 


Designation. 


S. 
R. 

L. 


9 . 7660909 




4.8780609 
4.4798652 

4.6894836 


Holston— High Knob. 
Big Knob— High Knob. 

Holston— Big Knob. 


a. c. log sin S =0.2339091 

log sin R = 9 . 3678952 

log LS =4.4409976 

\ogRL. -f- J 

a. c. logins }•■• =5-0731024 

log sin Z, = 9-5775I36 

log RS = 4.6506160 



The column marked correction by L. S. is for the correc- 
tions obtained in figure-adjustment. When it is not possible 
to make this adjustment, the error, after deducting spherical 
excess, must be distributed arbitrarily. If the angles are ap- 
proximately equal, one third the error should be applied to 



222 



GEODETIC OPERATIONS. 



each angle ; if not equal, the distribution should be propor- 
tional to the size of the angles. If one signal should be dim, 
or uncertain, it may be best to give to the angle between it and 
the other point the bulk of the error. Occasionally the angle 
deserving the greatest correction can be determined by ex- 
amining the individual readings. If they vary considerably, 
showing a wide range, the inference is that the average is 
somewhat uncertain, and that the principal source of error in 
the triangle is at this point. Such evidence as this, and the 
appearance of the signals from each other should have some 
weight in distributing the error. The most convenient form 
of blank for computation is to have three or four sets of the 
upper slip printed on the left side, and the same number of 
the lower, on the right side of a book. 



Names of Stations. 


Latitudes. 


£'=£-»" (i+ 


e 2 cos 2 L) cos Z - i sin i" sin 2 Z« //2 (i-f e 


2 cos 2 L) tan L. 


Holston 


Authority, 
Latitude (Z). . 

log A- (yds).. 


U. S. Geo. Survey. 
=36 27 27.41 


i sin 1" 

2 log sin Z. . . 

2 log Zl" 


.- 4.3845448 
.= 9.7058236 

. = 6.6969170 
.= 0.0018710 
.= 9.8685368 


in 

e3 
>> 

CO 

xn 
xn 

II 

High Knob 


b£ 

Bj 

Pk 

£6 
O 
O 

PQ 
6 

V 

C 

.2 


= 4.8780609 

= 8.4703976 


,0g ^sini"- 
loc u" 


— 3.3484585 


log(i -f- s' 2 cos 
log cos Z. . . . . 

log ist term. . 

ist term 

2d term 

SL 

Z 


i 1.)..= 0.0018710 

..( — )=: 9.8460032 


log tan Z 

log 2d term. . 

2d term 

Z + Z' 

£-{-£' 
2 


= 3-I963327 

..(+)= I57L57 
..(-)= 4-55 


.= 0.6576932 

• = 4"-55 

.=73 21 01.84 
.=36 40 30.92 


I567.O2 

/ // 
. .(+)= 26 07.02 
— 36 27 27.4I 


Latitude (Z'). 




=36 53 34-43 



GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 223 



Longitudes. 


Azimuths. 


Remarks. 


M' 


«"sinZ 

= M-\- — . 

cos L' 


Z' = Z ± 180 - (8M) sin — -*— . 
2 


Authority, 
Longitude 

log sin Z. . 

log u" . . . . 

log cos L' 
log (SAf). 

SAf. ..... 

At 

At' 


U. S. G. S. 
At = 82 04 3S.17 


U. S. G. S. 

Azimuth Z = 134 32 39.47 

180 




....(+) = 9.8529118 
= 3-3484585 


Z{+) 180 = 3143239.47 


3.2013703 
...(+)= 9-9029592 


- Z + Z' 
log sin ! ... = 9.7761771 

(-[-) — 3.2984111 


•■••(+) = 3-2984111 
(+) = 1987.98 

• • •(+) = 33 07.98 
= 82 04 38.17 




log 8Z (— ) = 3.0745S82 

SZ in seconds. . . = 1187.38 

sz (-)= 1947.38 

Z (+) 180 = 314 32 39.47 


= 823746.15 


Azimuth Z' = 314 12 52.09 



The latitude blank should occupy the left, and the longitude 
and azimuth the right side of a book. Two forms should be 
on each page, the second serving as a check computation, by 
determining the third point of the triangle from the other end 
of the base. For example: in triangle ABC, suppose L. M. 
Z. of A and B is known, C can be determined from A, and 
also from B. The average of these values is to be taken, to 
be used in connection with A or B in determining D, etc. . . . 



224 GEODETIC OPERATIONS. 



CHAPTER VIII. 

FIGURE OF THE EARTH. 

WITH the geographical positions of the termini of a line and 
its length known, it is possible to find an equivalent for its 
length along a meridian or a parallel, thus obtaining a value 
for a degree in that latitude. 

Assuming that the earth's meridian section is an ellipse of 
small ellipticity, we can develop a formula giving the length of 
an arc in terms of the terminal latitudes, the semi-axes, and 
ellipticity. Also the problem almost the converse, by which 
the values of the axes and ellipticity can be found. 

Let Z, L f , and /represent the terminal and middle latitudes 
of an arc whose amplitude is X ; a, b, and e, the semi-major, 
semi-minor axes of the meridian-section, and the ellipticity ; S, 
the length of the arc ; r, the radius vector, and 0, the geocen- 
tric latitude. 

The equation for the ellipse is 

x 2 f 

y. +.$ = *' (I ) 

x = r cos 6 , y = r sin 6, substituting in (i), 

r 2 cos 2 6 r 9 sin 2 6 _ 
a' + 3 a ~ I; 

,,. ., , 9 cos 2 6 sin 2 # I 

divide by r\ —j- + — n- = > (2) 



FIGURE OF THE EARTH. 225 

On page 207 we found 

P . sin 2 6 V sin a Z 



tan = -« tan Z, or 

from which sin 3 = 



a 8UU ^ U1 cos 2 0~ a 4 cos 8 Z' 
b* sin 2 Z cos 2 



a* cos 14 Z 
Substituting for cos 2 0, 1 — sin 2 0, and solving, we get 

• - _ ^ 4 sin 2 Z 

Sm ""^cos'Z + ^si^Z* 

By a similar process we get 

a # 4 cos 2 Z 

cos = -5 a r , 4 . a = ■ . 

<z cos Z -f- sin Z 

Placing these equivalents in (2), 

# 2 cos 2 Z-j- # 2 sin 2 Z 1 



a 4 cos"Z + ^sin f Z""^ 



(3) 



In the ellipse, & = a\i — f) 2 in which 6 is the ellipticity ; sub- 
stituting this in (3), 

^ 2 cos 2 Z + a 2 (i — *) 2 s?n 2 Z _ i_ 
a 1 cos 2 Z + a\i — *) 4 sin^Z ~~ ?' 

Dividing out <2 2 , and writing 1 — sin 2 Z for cos 2 Z, after reduc- 
tion, we have 

r 2 (i-26sin 2 Z+£ 2 sin 2 Z) 

= a\i - 4s sin 2 Z + 6e 2 sin 2 Z - 4s 3 sin 2 Z + £ 4 sin 2 Z) ; 

omitting terms involving powers of £ above the second, 
15 



226 GEODETIC OPERATIONS. 

r* = aXi-esin*L)\ 

r = a(i — e sin 2 Z). 



ds I dd* dr* dr 

dL=\/ r -dL> + dL" 



The formula for rectifying a polar curve is, 

dr 

dL 

dd 

--tj — I — 26 -\- 46 sin 2 Z. 

This is obtained by differentiating the equation 

a* tan = tf tan Z, or tan 6 = (1 — £) 2 tanZ; 

^ = *V(i-£sin 2 Z) 2 (i-2£ + 4£sin 2 Z) 2 + 4tf 2 £ 2 sin 2 Z cos 2 Z 

as 

= a{\ — 2«+3f sin 2 Z) ; 

omitting in the above all terms involving £ above the second 
power before extracting the square root. 



ds = a(i — 26-\-2> e sin 2 L)dL = #(i £ cos 2L)dL, 



placing sin 2 Z = £(1 — cos 2Z). 

Integrating the above between the limits Z and Z', we have 

s = #[(i — if) (Z — Z') — i f(sin 2Z — sin 2Z')], 

£ = #(i — e), from which £ = . Substituting this in the 

above equation, 



FIGURE OF THE EARTH. 227 

= (i±i) (Z _ r) - fcl)(sin 2Z - sin aZ') 

= #a + fy\ - #a - b) (sin Z cos L - sin Z' cos L'). . (4) 



L-L' = X, and /=.— i— . 2/=Z-f-Z'; 



sin A = sin (Z - Z') = sin Z cos Z' — sin Z' cos Z ; 
cos 2/ = cos (Z + Z') = cos Z cos Z' — sin Z sin Z'; 
sin X cos 2/ = sin Z cos Z cos 2 L — sin 2 Z . sin L cos Z' 

- sin I! cos 2 Z cos L + sin 2 Z' sin Z cos Z. 



Substituting in this 

sin 2 V = I — cos 2 Z', also sin 2 Z = I — cos 2 Z, 
it reduces to 

sin X cos 2/ = sin Z cos Z — sin Z' cos Z', 
which is the same as the last term in (4) ; therefore 

s _ i(a _|_ £)! _ |(* _ £) sin A cos 2/. (5) 

This requires a particular ellipsoid from which to obtain the 
value of a and b, but it gives a means of finding a and b, if all 
the other terms are known, which is the problem geodesy at- 



228 



GEODETIC OPERATIONS. 



tempts to solve. Suppose s, A, /, s' 9 A', I' be the lengths, am- 
plitudes, and mean latitudes of two arcs, we will have 

s = \{a -f- b)X — \{a — b) sin A cos 2/; 
s' = \{a + b)X' — %{a — b) sin A' cos 2/' ; 

solving for fe±S, and ( ^, 



# + b _ s' sin A. cos 2I — s sin A.' cos 2/' 
A sin A 7 cos 2I' — X' sin 'A cos 2/ 



2 
a — b 



s'\ - sX' 



3 ' A' sin A cos 2/ — A sin A 7 cos 2/" 



from which a, b, and £ can be found, s and s' are the distances 
between parallels, whereas in practice our lines make an angle 
with the meridian, so that its projection upon the meridian 
must be found. 

To find the effect of errors in the values s and s' upon a 
and b, we would differentiate the above equations, regarding s 
and s' only as variables. In the result the 
denominators would remain ; consequently 
the minimum error would occur when the 
denominator is a maximum, that is, when 
2/' = o, and / = go°, or when one arc is at 
the equator and the other near the pole. 

Let Pbe the pole of the spheroid, PM 
and iW two meridians passing through the 
points M and N, whose geographic and 
geocentric latitudes arc L, L', 6, and 6'. 
PM =go° - 0, and PN - 90 - 0', from 
FlG - 3 °- which NE = 6—6', which we will call x ; 

also the line NM= s, a known quantity. 




FIGURE OF THE EARTH. 229 

In the spherical triangle MPN, by Gauss's formulae, 

sin i(PiV- PM) cosiMPiV= sin \MN sin \{PMN - PNM)] 
cos i{PN- PM) cos \MPN= cos \MN sin \(PMN-\-PNM). 

Dividing the first by the second, 

tan*(/W- PM) = t™W# ^% PMJf+Pj9 S fi 

PMN = 180 - Z, />JWf = Z' - 180°. 

hence i(PMN - PNM ) = ^(360° - (Z + Z)) 

= 180 -i(Z + Z'), 

and i(PMN+ PNM) = \{Z - Z). 

Substituting these values, 

x • smi(Z4-Z') 

tan- = tan is- — ^p- — ri- 

2 * sm i(Z— Z) 



™ • r sin ±(Z -{- Z') , .r , j 

Placing ^ = . . (7 , ~r, we have tan - = h tan - ; writ- 

sin 2 \^* — ■" ) 22 

x s 

ing for tan - and tan - their developments, 

2 £ 



2 ' 24 ' 24O \2 ' 24 ' 240 / 

Solving this equation for # in terms of i", by approximation, 



23O GEODETIC OPERATIONS. 

we have for the first value x = s/i ; substituting this for x z and 
X s , we have, after transposing, 



x_shs^h_s^s^h L __s^ sh ski s' 1 - s*k* \ 

2 " 2 "" 24 ~ 24 ' 24O 240 * ' ' " 2 ' 2 \ 12 / 

+ 2A 120 ~r 



or 



^4 I .+ iJ I -^+i^ 1 -**>•••] 



for the second approximation ; and this value of x, substituted 
in the first equation, gives 



= tf[l + ~( ! - *") +iJo (l - *? (2 ~ 3 ^ ■ ■ • J 



for the (A) third approximation. 



If * = *n*(£+f) ,-*c=« *»■**+*> 



I-/* 2 = 



sini(Z'- Z)' ' sin 2 £(Z'- Z) 

sin 2 |(^- ^) - sin 2 i(Z+Z Q 
sin'^Z'-Z) 

|(i - cos (Z - Z) - j(i - cos (Z+Z) 
sirM(Z'-Z) 

— sin Z sin J? 7 



sin 2 £(Z' ~ Z)' 



FIGURE OF THE EARTH. 23 1 

Regarding the earth's meridian section as an ellipse, we 
know from the properties of an ellipse that 

x = a cos u } and y = b sin u y 

in which u is the eccentric angle, or reduced latitude. 
Differentiating the above, 

— dx — a sin udu, dy = b cos udu. 

If we consider this point, whose co-ordinates we have just 
written, to be in latitude Z, and an element of the elliptic 
curve to be ds, it will be the hypothenuse of a right triangle, 
in which 

— dx = ds sin L, and dy = ds cos L, 
or — dx — a sin udu = ds sin Z, dy — b cos udu = ds cos L. 
Dividing, 



a 

j- tan u = tan L, or a tan u — b tan Z, (i) 



a sin #dfo . N 

and ds — — : — 9 — • ( 2 ) 

sin L v J 



The value found for x in (A) was for a spherical surface ; to 
transform to an ellipsoid it will be necessary to pass to a dif- 



232 



GEODETIC OPERATIONS. 



ferential triangle on each. In the figure on page 228, suppose 
we call PNM a differential triangle on an ellipsoid, in which 
EN == dL, NM— do; and the angle PNM = a, then da cos a 
= dL. To convert dL into arc measure, we multiply it by the 
radius of curvature of the meridian, or 



ds cos a = RdL. 



(3) 



Likewise, if we conceive the same triangle to be on a sphere 
of radius a, then as will be the length of the arc MN, then 



adcr cos a — adu, or da cos a = du. 



(4) 



_ , , , s ds RdL 

Dividuig (4) by (3), -ft = -^ ; 



substituting in this R 



a{\ - S) dL (1 - ey 



(1 -^ 2 sin 2 Z)§' du ~ 1 -/cos 2 a 



also from (1), we find sin L = 



sin u 



(1 — e 1 cos 2 uf y 

ds_ a(i - e") (1 - ey 

da ~ (1 — e 1 sin 2 Lf ' I — ^ 2 cos 3 & 

a(i - ej 1 



[ 



sin u 



I — *' 



0(1 -ey 



1 — * cos u 



FIGURE OF THE EARTH. 

a{\ — ey I 



233 



1 — e* 



a(i —ey (1 -e 2 cos*u)i _ ,- 
~~ 1 - e* cc^~u ' (1 -"?)• " a 



e 1 cos 2 u y 



ds 



— — = a Vi — e 1 cos 2 u, 
dcr 



(5) 



which gives the relation between an infinitesimal length on a 
sphere to a corresponding length on an ellipsoid. 

To integrate this, Jordan takes a spherical triangle with 
sides equal to //, tt\ and cr, and angle opposite u x = a 1 ; then 



sin u = sin u cos cr 4- cos u sin a cos a , 



placing the serial value for cos cr and sin cr, 



sin u 



= sin u x [ 1 . . .) + (cr . . .) cos a 1 cos a 1 , 



omitting all powers of cr above the second. 
Squaring this equation, 

sin 2 u = sin 2 u\i — cr 2 ) -\- cr 2 cos 2 u 1 cos 2 a 1 

4- 2cr sin » x cos « x cos a 1 , 



For sin 2 u, write 1 — cos 2 u, then transpose, change signs, mul 



234 GEODETIC OPERATIONS. 

tiply by ^ 2 , subtract from I, and extract the square root; this 
gives 



V i — e l cos a u = i cos 2 u l + e'er sin u 1 cos u 1 cos a 1 

2 ' 

-) <x 2 (cos 2 & 1 cos 2 a 1 — sin 2 w 1 ). (6) 



Placing this in (5), and integrating with respect to da, we 
find 



= <n 1 cos u \ -J sin # cos & cos a 



e*a* 
-f- -^-(cos 2 w 1 cos 2 a 1 — sin 2 a 1 ). (7) 



This can be written, including e\ 



s ! <? . A 

— = <T I COS U 

a \ 2 / 

1 -| cr sin n x cos z/ 1 cos a: 1 -f- ^-cr 2 (cos 2 u l cos 2 «' — sin 2 ti l ) . 



If we place a: 1 = o, we have a = u — u\ then the last equa- 
tion becomes, after placing 5 for s, 



S e* 

— = (u — u l )(i cos 2 u l ) 

a v A 2 y 

1+ -(« — ^sin^cos^ + ^-O— ^(cos 2 */ 1 — sinV)J.(8) 



FIGURE OF THE EARTH, 235 

From sin u = sin u l \i J + cr cos u 1 cos a 1 , 

we get by transposition 

sin u — sin u 1 = cr cos & 1 cos a 1 sin u\ (9) 

But we had 

» — u l = x, or & = u 1 -f- ;r ; hence sin & = sin (u 1 -f- x). 

Developing this by Taylor's formula, 



x* 
sin m = sin u 1 -f- ;r cos a 1 sin a 1 ; (10) 



^ 2 
or sin & — sin u 1 = .rcos & 1 sin #' 

2 

= cr cos & 1 cos a 1 sin u\ (1 1) 



Solving this equation by approximation, 



x cos u 1 = cr cos u 1 cos a 1 , or ^r = cr cos a 1 . 



Substituting this in (11), 



cr' cr* 
x cos u 1 — — cos 2 a 1 sin u 1 = cr cos « x cos a 1 sin «*; 

2 2 



236 GEODETIC OPERATIONS. 

dividing by cos u 1 , 



x — — cos a 1 . tan u 1 = cr cos a 1 tan u\ 

2 2 



<7 2 O- 2 

x= (T cos a 1 tan u 1 4- — cos 8 a 1 tan & 1 

2 ' 2 



— cr cos a 1 tan u\i — cos 2 a 1 ) 

cr 2 
= <r cos a: 1 — — tan w 1 sin 2 a 1 = u — u 1 ; (12) 



substituting this in (8), 

— = (u — u l )ii cos 2 u 1 J I -| — cr sin u l cos & 1 cos a 1 

e'er' 2 ~1 

_| (_ s i n 2 u x _|_ 2 cos 2 7/ 1 cos 2 a 1 -j- sin 2 u l cos 2 a 1 ) . (13) 

12 _l 

Dividing this equation by (7), we have 

5 u-ttT e*o*. . .■.,,-., .tl 

— = I (—3 sin u -\-2 sin & -+- sin # cos a 

= I (— sin 2 & 1 -f- sin 2 &\i — sin 2 a: 1 ) 

in 2 a). (14) 



^ • a 1 

I — sin & sin 



12 



FIGURE OF THE EARTH. 2tf 

If we had taken u 1 as the unknown side in our spherical tri- 
angle, giving 

sin u 1 = sin u cos a -\- cos u sin cr cos a, 

we would have found, by pursuing a course similar to the 
above 



5 u— u 

s ~ 



I sin u sin or J, 

cr V 12 /' 



from which we could obtain 

sin 3 u 1 sin a a 1 = sin a # sin 2 a, or sin & 1 sin a?=z sin & sin or ; 

hence we can involve both the direct and reverse azimuth as 
well as the terminal latitudes by writing for sin 2 u 1 sin 2 a 1 , sin 
u 1 sin a 1 sin u sin a, so that (14) will become 



S _u — 
s 



- u 1 / e'er 2 \ 

1 1 sin u x sin a x sin u sin a), (15) 



Resuming the former notation, we will put a = Z, and a 1 = 

~, „ o , 1 ,, s'm Z sin Z 1 

Z — 180 ; also, remembering that 1 — h = 7 ~*TTW — tv 

sin 2\~ ~~~ ) 
we can for — sin Z sin Z 1 write sin a J(Z x — Z) (1 — #*), and 
substitute in (15) the value of u — u 1 in (A), which gives 

— = # 1 1 sin & sin 2/ sin Z sin Z) 

s \ 12 y 

[i-g (1 -h>) + ^(1-^2-3^]. (16) 



238 GEODETIC OPERATIONS. 

This still involves s and 0", so there is needed a relation be- 
tween them. To attain this we take equation (7), using only 
two terms, 

s ! <? . A i e * A 

— = <n I • COS « I = 0"l I cos u cos u\. 

Squaring this, and omitting terms in e\ 

- T = 0- 2 (i — ^ 2 cos m cos & 1 ), 

or 2 — -tt; i IT- ( x 7) 

<z (1 — e cos ucosu) v /y 

Writing / 2 = 1 + e* cos # cos u\ 

(16) becomes 

r eYf . r . . ■ „ t , „ x 

5 = $/* (1 — — — 5 sin a sin & sin Z sin Z ) 

sY( sin Z sin Z 1 \ _ j 4 / sin Z sin Z 1 ~| 

^ + I2# 2 \sin 2 HZ 1 - Z)l ~ 240a' sin 2 J(Z 2 - Z~/ 2 ~~ 3 ^ 'J' 

sin i(Z+Z>) 



in which A = 



sin \[Z l — Z)' 



This is substantially the same formula as given by Bessel in 
Astronomische Nachrichten, No. 331, pp. 309-10, except Z 1 is 
within the polar triangle, which gives 

_ cosK^ + ^ x ) 
H ~ cos i-(Z - Z 1 )' 



FIGURE OF THE EARTH. 239 

sin Z sin Z x 

cos 2 i(Z x — Z) 

or approximately sin Z sin Z\ 

Then, writing 

/ 2 = 1 -f- e* cos # cos u\ and # a = 1 -)- ^ 2 cos (« -f" u l ), 

Bessel's formula becomes 

L 12W/ cos 2 ^(Z— Z ) 

I /j^V sin Z sin iT 1 "1 

+ 245 W cos 2 i(Z~-=^) ^ 2 ~~ 3/ * )J- 

In both of these formulae it is to be remembered that 



tan u = Vi — e* tan L, and tan u 1 = Vi — e* tan L 1 . 

Also, if the line deviates but little from the meridian, the 
first term will be sufficient. 

When a long arc has been measured, it has been found best 
to divide it into several sections, from each of which data can 
be obtained for finding the axes of the earth, and the ellipticity. 
When these arcs are small, the method given on page 228 will 
give fair results. But Clarke's solution is perhaps the best ; it 
is, in the main, as follows: 

Let R, x, and y be the radius of curvature of an ellipse, and 

x 1 y 2 
co-ordinates of the point whose latitude is Z, then — 3 -j- 5 = I ; 

but we have shown that 



x = a cos u> tan u = Vi — e 1 tan L, 



240 GEODETIC OPERATIONS. 

from which 

x = a cos Z(i — e 1 sin 2 Z) - fr, 

also 

jj/ = # sin # = # sin Z(i ~ - 2 ) (i — ^ 2 sin 2 Z) ~ i ; 

^=^(i-^ 2 )(i-^sin 2 Z)-i 

Expanding, and neglecting e\ 

x = 4(i + tf + ^) cos Z-(^ 2 + xh/Jcos 3^+ rfc^cos $£] ; 

j, - 4^ _ |^_ ^) sin Z-(i^ 2 - ^O sin 3 Z+ T f ^ 4 sin 5Z] ; 

.£ = 0(1 _ ^ 2 ) (I -f 3^ s { n * X _|_ JJ>^ sin 4 L y 

Substituting 1 — cos 2 Z for sin Z 2 , 

R=a[i- \? - 7 V 4 - (V + A' 4 ) cos 2Z + ^ 4 cos 4 Z]. 

Writing ^ = 0(1 - i^ 2 - ¥ V 4 )> •# = - ^(l^ 2 + sV 4 )> C = Jfttf/, 

we have R — A -{- 2B cos 2L -\- 2C cos 4L, (B) 

This is an ellipse if 5^ a = 6^4 £7. 

Now, if 5 be the length of an arc of an elliptic meridian, it 
was shown on page 231 that 

jc - r -J jc a sin ud u 

dS sin Z = a sin uau, as = : — j — . 

sinZ 



FIGURE OF THE EARTH. 24 1 

From a tan u = b tan L, we found 



dii_ Vi -e 2 

dL - 1 - e* sin a Z' 



ds a sin « y 1 

therefore 



dL sin Z(i — e* sin 2 Z)' 
But from the preceding relation 



sin L V 1 — e* 
sin u = Vi - ? sin 2 L 



Substituting this, we have 

dS ail — e 3 ) _, . , „ r , „ 

-j T = , V-ttv =R — A + 2B cos 2L -f 2C cos 4Z • 

dL (1 — e sin Z)* ' ^ 

by integration, 

5 == AL -f- i? sin 2Z -} JC sin 4Z -|- a constant. 

If Z be the mean latitude of an arc whose amplitude is A, 
and the above expression be integrated between the limits 
L — \"k and L -f- -JA, we will obtain 

5 = Ak -f- 2i? cos 2Z sin A -{- C cos 4Z sin 2A. (19) 

In this A, B, and C are the only unknown quantities, so that 
if S, L, and A be free from errors, three equations would be suf- 
ficient for determining A, B, or C, and, consequently, <z, e, and 
16 



242 GEODETIC OPERATIONS. 

b. But every arc is affected with an error, in length as well as 
middle and terminal latitudes, so that from a number of dis- 
cordant results we must find the most probable values for A, 
B, and C by the principles of least squares. 

Suppose the terminal latitudes have a small error in each of 
x l and x,\ so that the amplitude would be X-\- x^ — x v and 
the latitudes L — \X -\-x v and L — \\-\- x t \ 

Placing these corrected values in (19), 

5 = A(X -f x? - x 1 ) + 2Bcos 2L sin (A + x, 1 - x,) 

-f- C cos 4L sin 2 (A -\- X* — x x ). (20) 

In expanding this we treat x^ — x 1 as a single term, and 
being small, cos (x^— x y ) = 1, and sin (x* — x x ) = X* — x v so 



sin {X-\-x^ — xj = sin X -f- cos X(x* — x x ) ; 

sin 2(X-{-x 1 1 —x 1 )= 2 sin [X -\- (x, 1 — xj] cos [A. -|- (x* — xj] 

= 2[sin A+cosA^ 1 — .arjjfcos X— sin ^{x^—x^j\ 
= sin 2A -|- 2(;r 1 1 — ^) cos 2 A ; 

substituting these expressions in (20), 

S= A(X-\-x 1 1 — x,) -\- 2B cos 2Z[sin A -j- cos X(x, 1 — x,)] 

-f- Ccos 4^[sin 2A -J- 2(^r 1 1 — *,) cos 2A]. 

Solving for x* — x, 

(x, 1 — x,) (A + 2i?cos 2Z, cos A 2 Ccos 4Z cos 2A) = 5 — AX 

— 2^5 cos 2L sin A — CC0S4Z sin 2A. (21) 

If we write ^4 -f- 2B cos A cos 2L 2C cos 4Z cos 2 A = — , (21) 
will reduce to 



FIGURE OF THE EARTH. 243 



\ 1 -*>=\a- x )> x - 



2BM . . 

—j- sin A cos 2L 



—j sin 2A cos 4Z. (22) 



Expressing x*, x v and A in seconds — we approximate the 
length of a second of latitude by assuming the average radius 
of curvature to be 20855500 ft. — we must write 

v — 20855500 sin 1". 

Then we assume three auxiliary quantities, u, v, and Z, and 
place 



1 + 



!L_) 



A 20855500V ' iooooV 

2B _ 1 v 
A ~~ 200 ' 1 0000' 

C Z 



A 1 0000' 
Substituting these, (22) becomes 



1 , . . 5 , Su _ , sin A cos 2L 

-{X x — XJ = - -| A -A : 77- 

}A> 1J V ' IOOOOU ' 200 sin I 



sin A cos 2Lv sin 2A cos 4LZ 
1 0000 sin \" Tnnnn cin T " 



Again we assume 



,'S .. , sin A cos 2L\ Sjj. 



v * 200 sin I" /' iooool' 



244 GEODETIC OPERATIONS. 

_ sin X cos 2Lp sin 2\ cos ^Ljj. 

ioooo sin i'"' i oooo sin i" ' 

jj. = I -f- ^-J-g- cos A cos 2Z. 
Then (23) can be written 

x* — x x = m -f- au + ^ -f" <:Z, 
or ^"j 1 = ^ -(- M + ## + bv + ^ (24) 

For each arc or partial arc we will have an equation like (24), 
which is to be solved by the principles of least squares, by 
making the sum of the squares of the errors a minimum ; then 
equating the differential coefficients of the symbolic errors 
with respect to u, v, z, x t \ etc., to zero, there will be as many 
equations as there are unknown quantities to be solved by al- 
gebraic methods. Knowing u, v, and z, we find A, B, and C, 
which substituted in (B) give R. 

To determine the axes and ellipticity, we take the equations 
on page 240 and find that the coefficient of cos L = (A — B), 
of cos 3Z, = i(B — C), and of cos 5Z, = \C', also, of sin L = 
A -{-B, of sin 3Z = i(B -\-C), and of sin 5Z = \C. By making 
these substitutions, we have 

x—{A- B) cos L + ftB— C) cos 3L +%Ccos 5Z ; (25) 
y = (A + B) sin L + i(B + C ) sin 3L + \C sin 5Z. (26) 

But on page 231, 

x = —fa sin L dL, and y ' =fb cos L dL. 



FIGURE OF THE EARTH. 245 

So (25) and (26) are the values of these integrals, which if in- 
tegrated between the limits L = o and L — 90 , will give 
the semi-axes, 



[> = A + B-i(B + C) + jrC = A + %B-£rC; (28) 
V 8B( aB\ . . 

? = 1 - -* = - ^i + jzJ' a PP roximatel y- 

If these values be substituted in (25) and (26), we would have 

S = (A-B-^C + l -?)cosL 

+ (i i? -£) COS3Z + 5 COS5Z; ^ 



f = [A+B-^C+ l -§jsinL 

+ (^-lfi) sin 3^ + S s!n 5 Z - (3°) 



(29) and (30) are the values of the co-ordinates of a point in 
an elliptic curve whose axes are a and b, while (25) and (26) 
are the co-ordinates of a point in the actual curve. The dif- 
ference between the two will be the deviation of the actual 
from the elliptic curve at any point. 



' 1 = { C ~ HXls cos l " \ cos lL + \ cos sL )' (3I) 



246 



GEODETIC OPERATIONS. 



y-f ={ C -^a)[tc sinZ + \ sin 3^+ ^sin sL). (32) 



Suppose P be the point on the 

elliptic curve in latitude Z, and Q 

the point on the actual curve in the 

same latitude. P and Q will coin- 

c /? 2 
cide when C — ^-,- — o, for this will 
6A 

reduce (31) and (32) to x — x' = o, 

y — y 1 — o, and will differ from one 

another as C — j-v- changes from 

a zero value. 

If we take PS an infinitesimal 
distance on the elliptic curve, and QS a corresponding length 
along the normal, we will have 




PT = y-y\ 



TQ 



x\ 



QS= QU+SU= QU+PF 

== (x — x 1 ) cos L + (y — y 1 ) sin Z, 



or dR — (x — x') cos L -\- L(y — y 1 ) sin L. 



(33) 



PS= VU= TU- TV 

= — (x — x 1 ) sin L -\- (y — y 1 ) cos L, 



or dS = — [x — x') sin L -f- (y — y l ) cos Z. 



(34) 



FIGURE OF THE EARTH. 247 

Substituting in these equations the values of {x — x 1 ) and 
{y — y l ) from (31) and (32), we find 

^ = r 5 ( c -S) sin4Z ' 

Clarke's values of a and b of 1866 would eive 



showing but a slight deviation of a meridian section from an 
ellipse. 

The Anglo-French arc places the actual curve 3.6 feet under 
the ellipse in latitude 58 , and 18.9 feet above in latitude 44 ; 
while the Indian arc places it 19.6 feet under, in latitude 14 , 
and 9.3 feet above, in latitude 26 . 

The amplitude of an arc depending upon the latitude deter- 
minations of its extremities is subject to an error from local 
deflection. In some cases, at least a portion of these errors 
can be corrected by computing the effects of attraction upon a 
physical hypothesis; but in the main they are best treated 
as accidental, and the figure of the earth determined by the 
principle of least squares, in which the sum of the squares 
of all errors shall be a minimum. 

This was suggested by Walbeck in 18 19, continued by 
Schmidt in 1829, and perfected by Bessel in 1837. 

Laplace in 1822, published the second volume of Me'canique 
Celeste, in which he discussed the figure of the earth, using 
seven arcs : the Peruvian, Lacaille's Cape of Good Hope arc, 
Mason and Dixon's, Boscovich's Italian, Delambre and Me- 
chain's, Maupertuis' Lapland arc, and Liesganig's Austrian arc. 



248 GEODETIC OPERATIONS. 

The second is unreliable, from an erroneously assumed cor- 
rection for local attraction which shortened the arc by 9" too 
much. The third was a measured arc, and not comparable 
with a trigonometric one. And no confidence is now placed 
in either the fourth or the last. 

Bowditch, in his translation of the above-named work, con- 
siders only the Peru and France arcs, and adds, those of Eng- 
land and India as completed in 1832. His conclusion is: 

" It appears that this strictly elliptical form of the meridian 
is more conformable to these observations than the irregular 
figure obtained by Mr. Airy's calculation." 

Sir George Airy published in the Encyclopedia Metropoli- 
tana, under the heading " Figure of the Earth," in 1830, a dis- 
cussion of fourteen meridian arcs and four arcs of parallel. 
In 1841, Bessel gave to the public the results of his laborious 
investigation of ten meridian arcs, having a total amplitude of 
5o°.5, and embracing thirty-eight latitude stations. The re- 
sult gave an elliptic meridian, and the elements then published 
are still known as those of Bessel's spheroid. 

In 1858, in the "Account of the Principal Triangulation of 
Great Britain and Ireland," Captain Clarke gives a most elabo- 
rate discussion of eight arcs, having a total amplitude of 78 
36', and embracing sixty-six latitude stations. 

Again, in 1880, he revised his previous computations, using 
corrected positions from which slightly different results were 
obtained. 

Mr. Schott discussed the combination of three American 
arcs of meridian for determining the figure of the earth con- 
sidered as a spheroid. He used the Pamlico-Chesapeake, Nan- 
tucket, and Peruvian, having a total amplitude of 11° oi' 12", 
and embracing twenty-three latitudes. The conclusion de- 
duced by Mr. Schott is : " The result from the combination of 
the three American arcs is the preference it gives to Clarke's 
spheroid over that of Bessel." 



FIGURE OF THE EARTH. 



249 



TABLE GIVING THE ELLIPTICITY AND LENGTH OF A QUAD- 
RANT ON THE SPHEROIDAL HYPOTHESIS. 



Date. 



1819 
1830 
1830 
1S41 

1856 
1863 
1S66 
iS63 
1S72 
1877 
1S78 
1SS0 



Authority. 



Walbeck 
Schmidt. 

Airy 

Bessel. . 
Clarke. . 
Pratt. . . 
Clarke. 
Fischer. 
Listing. . 
Schott. . 
Jordan. . 
Clarke. . 



Ellipticity. 



302.8 

297.5 

299-3 

299.2 

298.I 

295.3 

295 

288.5 

289 

305.5 

286.5 

293.5 



Quadrant in Metres. 



10 000 
IO OOO 
IO OOO 
IO OOO 
IO OOI 
IO OOI 
10 OOI 
10 001 
10 000 
10 002 
10 000 
10 001 



268 

075 
976 
856 

515 
924 

888 

714 
218 
232 

681 
869 



Data for the Figure of 
the Earth. 



Bessel, 1841. 



Clarke, 1866. 



Equatorial radius, a. . 6 377 397. 2M 6 378 206. 4M 
Polar semi-axis, b. . . . 6 356 079 
a — b 



Compression, 

Mean length of a deg. 



1 : 299.15 
in 120. 6M 



6356583- 
1 : 294.98 
in 132. 1 



Coast Survey, 
1877. 



Clarke, 1880. 



6378054.3M6378248.5M 

6 341 895. 6M 



6 357i7:> 
1 : 305. 4S 
in 135.9 



1 : 293.5 
in 131. 8 



The value of the ellipticity as deduced by pendulum-obser- 
vations in accordance with Clairaut's theorem is I : 292.2, be- 
ing almost the same as that obtained from geodetic measure- 
ments. 

Clarke's length of the quadrant would give for the metre 
39.377786 inches, whereas the legal length is 39.370432 inches, 
or .0073 inch too short. 



LITERATURE OF THE FIGURE OF THE EARTH. 

Pratt, A Treatise on Attractions, Laplace's Functions, and 
the Figure of the Earth. London, 1861. 

-Roberts, Figure of the Earth. Van Nostrand 's Engineering 
Magazine, vol. xxxii., pp. 228-242. 



250 GEODETIC OPERATIONS. 

Merriman, Figure of the Earth. New York, 1881. 

U. S. Coast Survey Report for 1868, pp. 147-153. 

U. S. Coast and Geodetic Survey Report for 1877, PP- 84-95. 

Clarke, Geodesy, pp. 302-322. London, 1880. 

Laplace, Mecanique Celeste. Bowditch's Translation, vol. 
ii., pp. 358-485. Boston, 1830. 

Ordnance Survey, Account of Principal Triangulation, pp. 
733-782. London, 1858. 

Bruns, Die Figur der Erde. Berlin, 1878. 

Baeyer, Grosse und Figur der Erde. Berlin, 1861. 

Jordan, Handbuch der Vermessungskunde, vol. ii., pp. 377- 
463. Stuttgart, 1878. 



FORMULA AND FACTORS 



FORMULA AND FACTORS. 2^ 



TRIGONOMETRIC EXPRESSIONS. 



sin 2 a -f- cos 2 a = i ; 



sin a = Vi — cos 2 a 

_ cos a 
~ cot a 



Vi + cot 2 a 
= cos « tan # 



2 sin i# cos ^ 







cosec a 




cos tf 


= 


sin <z 




tan a 






= 


sin # cot a 




Vi — sin 


2^, 




= 


1—2 sin 2 


i#. 






I 





sec a 



sin # 

tan # = 

cos # 



i 
cot a 



254 GEODETIC OPERATIONS. 



sin a 







Vl — sin 2 a 




— 


i — cos 20 
sin 20 




j\ — cos 2a 




y I + cos 2a 


cot a 


= 


I 


tan a" 


sec a 




I 



cosec a 



cos a 



sin « 



versin = i — cos a = 2 sin 2 \a. 



chord a = 2 sin -J0. 

sin (0 ± £) — sin a cos £ ± cos sin £ ; 

cos (a ± b) = cos a cos 3 =F sin sin £. 

tan a ± tan £ 
tan (« ± *) = T^TTteO" 

, yN cot. a cot # =F i 
cot(«±*) = cot j ±cota - 

sin 20 = 2 sin cos 0. 



cos 20 = cos 3 — sin 2 

= 2 cos 2 0—1 = 1—2 sin 2 0. 



FORMULAE AND FACTORS. 255 

2 cos 2 \a = 1 + cos a. 

2 sin 2 \a — 1 — cos #. 
1 — cos a 



tan 2 \a 



I -|- cos # 



sin a ± sin £ = 2 sin J(<z ± £) cos £(# =F £). 
cos a + cos £ = 2 cos J(« -j- 3) cos f (« — £). 
cos a — cos ^ '= 2 sin -J(tf -f- ^) sin i(^ — #)• 
sin 2 a — sin 2 <£ = sin (a -f- #) sin (<z — 3). 
cos 2 a — sin 2 # = cos (a -\- b) cos {a — b). 

sin 2;tr = 2 sin ^r cos ^. 

sin 3^tr = 3 sin x — 4 sin 3 ;r. 



sin 43; = (4 sin ^r — 8 sin 3 ;r) Vi — sin 2 x. 



cos $x = 4 cos 3 ^ — 3 cos ;r. 

cos 4jt = 8 cos 4 x — 8 cos 2 x -\- 1. 



tan 2^r 


= 


2 tan 4r 




1 — tan 2 x 




tan 


3* 


= 


3 tan x — tan 3 .#• 




1 — 3 tan 2 x 




tan 


4* 


— 


4 tan x — 4 tan 3 


# 



256 GEODETIC OPERATIONS. 



TRIGONOMETRIC SERIES. 



xr t X 

sin x = x j- 



2.3 ■' 2.3.4.5 2. ..7 
X* , X* X* , 

cos ^r = 1 ■ -2 + • • • 

2 '2.3.4 2 . . . 6 ' 

* 3 2JF 5 , I7^ 7 

tan x = x + — + — - + , a 

3 3-5 • 5 • 7 

I „r ^r 3 24T 5 

cot x — - 



* 3 3.5 3-5-7 

^ 2 $x* 6ix* 

sec * = 1 + -- + 



2 3 3 ' 2\ 3 2 .5*"* 

cosec .*■ = — z — r- -\ — : — • \- . 

x T 2 . 3 ^ 2 3 . 3 2 5 ^ 2 4 . 3 3 . 5 . 7 ~ 

. sin 3 ^r 3 sin 5 x , 3.5 sin 7 jf 

arc ;tr = sin x H H — ^ . . . 

1 2.3^2.4.5^ 2.4.6.7 

== tan x — \ tan 3 x + \ tan 5 x . . . . 
For very small angles Maskelyne's series is best. 

sin x = ^ycos x -\- . . . = x\ 1 — ->- J. 

3 / — ^— , / , x* , x*\ 
tan^r = x vsec x-\- . . . = ;d I -j 1 — J. 

BINOMIAL, EXPONENTIAL, AND LOGARITHMIC SERIES. 
(a-\-b) n = «*+ na n ~*b -f- ~ > a n - 2 F . . . b n . 

X X* X* 

a x — 1 + l°S — h log 8 h log 3 # . 

1 ta 1 ' & 1 . 2 ' s 2.3 



FORMULAE AND FACTORS. 
I I 



257 



log (*+!) = 2M[^ ¥i + ^—— + J ^^ . 

M = nodulus == 0.4342945. 
log M = 9-6377343. 



■). 



CONVERSION OF METRES TO FEET. 

Metres X 3.280869 = feet, or to log of metres add 0.5159S89 
" X 1.093623 = yards, 0.0388676 

11 X 0.000621377 = mile, " 6.7933550—10. 

1 toise = 76.734402 inches = 864 lines. 

1 Prussian foot = 139.13 lines. 
1 klafter = 840.76134 lines. 

The toise is that of Peru, which is a standard at 13 R. 



CONVERSION TABLES. 

METRES INTO YARDS. 

I metre = 1.093623 yards. 



Metres. 


Yards. 


Metres. 


Yards. 


Metres. 


Yards. 


IOO OOO 


109 362.3 


3 OOO 


3 280.87 


60 


65.617 


90 OOO 


98 426.I 


2 OOO 


2 187.25 


50 


54.681 


80 OOO 


87 489.8 


I OOO 


1 093.62 


40 


43-745 


70 OOO 


76 553.6 


900 


984.26 


30 


32.809 


60 OOO 


65 617.4 


800 


874.90 


20 


21.872 


50 OOO 


54 681.2 


700 


765.54 


10 


10.936 


40 OOO 


43 744-9 


600 


656.17 


9 


9-843 


30 OOO 


32 808.7 


500 


546.81 


8 


8.749 


20 OOO 


21 872.5 


400 


437-45 


7 


7-655 


10 000 


10 036.2 


300 


328.09 


6 


6.562 


9 000 


9 842.61 


200 


218.72 


5 


5.468 


8 000 


8 748.98 


IOO 


^09.36 


4 


4-374 


7 000 


7 655.36 


90 


9S.426 


3 


3.281 


6 000 


6 561.74 


80 


87.490 


2 


2.187 


5 000 


5 468.12 


70 


76.554 


I 


1.094 


4 000 


4 374-49 











17 



253 



GEODETIC OPERATIONS. 



Conversion Tables — Continued. 

YARDS INTO METRES. 

I yard = 0.914392 metre. 



Yards. 


Metres. 


Yards. 


Metres. 


Yards. 


Metres. 


100 000 


91 439.2 


3 000 


2 743-18 


60 


54.864 


90 OOO 


82 295 


3 


2 000 


I 828.78 


50 


45.720 


80 OOO 


73 151 


3 


1 000 


914-39 


40 


36.576 


70 OOO 


64 007 


4 


900 


822.95 


30 


27.432 


60 OOO 


54 863 


5 


800 


731.51 


20 


18.288 


50 OOO 


45 719 


6 


700 


640 . 07 


IO 


9.144 


40 OOO 


36 575 


7 


600 


548.64 


9 


8.230 


30 OOO 


27 43i 


8 


500 


457-20 


8 


7.315 


20 OOO 


iS 287 


8 


400 


365-76 


7 


6.401 


IO OOO 


9 143 


9 


300 


274-32 


6 


5.486 


9 OOO 


■8 229 


53 


200 


182.88 


5 


4-572 


8 000 


7 3i5 


13 


100 


91.44 


4 


3.658 


7 000 


6 400 


74 


90 


82. 295 


3 


2.743 


6 000 


5 486 


35 


80 


73-I5I 


2 


1.S29 


5 000 


4 571 


96 


70 


64 . 007 


1 


O.914 


4 000 


3 657 


57 


1 









METRES INTO STATUTE AND NAUTICAL MILES. 

I metre = 0.00062138 statute mile. 
1 metre = o 00053959 nautical mile. 



Metres. 


Statute Miles. 


Nautical Miles. 


Metres. 


Statute Mi!es. 


Nautical Miles. 


IOO OOO 


62.T38 


53-959 


900 


0.559 


O.486 


90 OOO 


55-924 


48.563 


800 


0-497 


0.432 


80 OOO 


49.710 


43.167 


700 


0.435 


0.378 


70 OOO 


43.496 


37-772 


600 


0-373 


0.324 


60 OOO 


37-283 


32.376 


500 


O.311 


O.270 


50 OOO 


31.069 


26.980 


400 


O.249 


O.216 


40 OOO 


24-855 


21.584 


300 


O.1S6 


0.162 


30 OOO 


1S.641 


16.188 


20O 


O.124 


O.I08 


20 OOO 


12.428 


10.792 


IOO 


O.062 


O.054 


10 OOO 


6.214 


5-396 


90 


O.056 


O.049 


9 OOO 


5-592 


4-856 


80 


O.05O 


O.043 


8 OOO 


4.971 


4-317 


70 


O.043 


O.038 


7 000 


4-35Q 


3-777 


60 


O.037 


O.032 


6 000 


3-728 


3-233 


50 


O.031 


O.027 


5 000 


3.107 


2.698 


40 


O.025 


0.022 


4 000 


2.486 


2.158 


30 


O.OI9 


O.OI6 


3 000 


1.864 


1. 619 


20 


O.OI2 


O.OII 


2 000 


1.243 


1.079 


10 


O.006 


O.OO5 


1 000 


0.621 


540 









FORMULA AND FACTORS. 



259 



Conversion Tables — Continued. 

STATUTE AND NAUTICAL MILES INTO METRES. 

i statute mile = 1609.330 metres. 
1 nautical mile = 1853.248 metres. 



Miles. 


Metres in 


Metres in 


Miles. 


Metres in 


Metres in 


Statute Miles. 


Nautical Miles. 


Statute Miles. 


Nautical Miles. 


100 


160 933.O 


185 324.8 


•9 


I 448.40 


I 667.92 


90 


144 839.7 


166 792.3 


.8 


I 287.46 


I 482.60 


80 


128 746.4 


148 259.8 


•7 


I 126.53 


I 297.27 


70 


112 653. I 


I29 727.4 


.6 


965.60 


1 in. 95 


60 


q6 559-8 


III 194.9 


-5 


804.67 


926.62 


50 


80 466.5 


92 662.4 


•4 


643 • 73 


741 30 


40 


64 373-2 


74 129-9 


•3 


482.80 


555-97 


30 


48 279.9 


55 597-4 


.2 


321.87 


370.65 


20 


32 186.6 


37 065.0 


.1 


160.93 


185.32 


IO 


16 093.3 


18 532.5 


.09 


144.84 


166.79 


9 


14 483-97 


16 679.23 


.08 


128.75 


148.26 


8 


12 874.64 


14 825.98 


.07 


112.65 


129.73 


7 


11 265.31 


12 972.74 


.06 


96.56 


in. 19 


6 


9 655-9 3 


11 119.49 


.05 


80.47 


92.66 


5 


8 046.65 


9 266.24 


.04 


64.37 


74-13 


4 


6 437-32 


7 412.99 


•03 


48.28 


55.60 


3 


4 827.99 


5 559-74 


.02 


32.19 


37-o6 


2 


3 218.66 


3 706.50 


.01 


16.09 


18.53 


1 


1 609.33 


1 853-25 






. ... .1 



Major semi-axis = a, minor semi-axis = b, ellipticity = e 
a — b 



Bessel, a = 6377397 . i$M, log = 7.8046434637 ; 
b — 6356078 . g6M, log = 6.8031892839 ; 
e = 



log = 6.8046985352; 
log = 6.8032237974 ; 



Clarke, a = 6378206 . 4M, 
b = 6356583 . BM f 



26o 



GEODETIC OPERATIONS. 



CONSTANTS AND THEIR LOGARITHMS. 



Ratio of circum. to diameter, 



Number. 
7T 3.I415926 
27t 6.2831853 
7T S 9,8696044 

Vn 17724538 

Number of degrees in circum., 360 

Number of minutes in circum., 21600 

Number of seconds in circum., 1296000 

Degrees in arc equal radius, 57°« 2 95779 

Minutes in arc equal radius, 3437 .7467 
Seconds in arc equal radius, 206264 .806 



Length of arc of 1 degree, 
Length of arc of I minute, 
Length of arc of I second, 

Naperian base, 

sin 1" 
i sin 1" 



.OI74533 
.0002909 
.00000485 



Log. 
O.4971499 
O.7981799 
O.9942997 
O.2485749 

2.5563025 
4-3344538 

6.1 126050 

1. 7581226 

3-5362739 
5.3144251 

8.2418774 — 10 
6.4637261 — 10 
4.6855749- 10 



2.7 1 828 1 8 0.4342945 

4.6855749 
4.3845449 



N is the normal produced to the minor axis. R is the radius 
of curvature in the meridian. Radius of curvature of the 
parallel is equal to TV cos L. 

The following tables are based upon Clarke's spheroid of 
1866, and were computed in 1882. Since then similar tables 
have been published by the Geodetic Survey, with which the 
appended have been compared. 



FORMULA AND FACTORS. 



26l 





N = 

(1 - * 2 sin 2 ZH 


K «(« - ^) 




Lat. 


(1 - e 2 sin* L)f 


Log (i + <?2 COS 2 L). 




Log N. 


LogK. 




24°oo' 


6. 804941 S 


6.802479O s 


0.0024628 


IO 


9450 


4884 


4566 


20 


9481 


4981 


4500 


30 


9512 


5076 


4436 


40 


9545 


5174 


4371 


50 


9577 


5270 


4307 


25 OO 


9612 


5370 


4242 


IO 


9645 


5470 


4175 


20 


9677 


5509 


4108 


30 


9711 


5667 


4044 


40 


9744 


5768 


3976 


50 


9777 


5869 


390S 


26 00 


9812- 


5968 


3841 


10 


9846 


6070 


3774 


20 


9880 


6173 


3706 


30 


9915 


6276 


3639 


40 


9948 


6379 


3569 


50 


9981 


6482 


3499 


27 00 


6.8050017 


6585 


3432 


10 


0051 


66S8 


3363 


20 


0086 


6794 


3292 


30 


0120 


6899 


3221 


40 


0156 


7006 


3150 


50 


0191 


7111 


30S0 


28 00 


0227 


7216 


3011 


10 


0263 


7322 


2941 


20 


0299 


7429 


2870 


30 


0334 


7537 


2797 


40 


0371 


7644 


2727 


SO 


0407 


7752 


2655 


29 OO 


0444 


7862 


2582 


IO 


0480 


797i 


2509 


20 


0517 


8081 


2436 


30 


0555 


8187 


2368 


40 


0591 


8296 


2295 


50 


0628 


8408 


2220 


30 OO 


0664 


8524 


2140 


10 


0700 


8636 


2064 


20 


0738 


8747 


1991 


30 


0776 


8858 


1918 


40 


0813 


8972 


1841 


50 


0849 


9084 


1765 


31 OO 


0891 


9198 


1693 


IO 


0928 


9310 


1618 


20 


0976 


9426 


i55o 


30 


1014 


954o 


1474 


40 


1054 


9654 


1400 


50 


1089 


9769 


1320 



262 



GEODETIC OPERATIONS. 





N = 

(x - e* sm» L)i 


a «0 - * 2 ) 




Lat. 


(1 - *a sins /,)§■ 


Log (i+^ 2 cos 2 Z). 




LogN. 


LogX. 




32°oo' 


6.8051128 


6.8029885 


O.OO21243 


IO 


II66 


6.8030002 


1164 


20 


1205 


0117 


1088 . 


30 


1244 


0232 


IOI2 


40 


12S3 


0349 


0934 


50 


1322 


0466 


0856 


33 00 


1351 


0583 


O768 


10 


139O 


0700 


069O 


20 


1429 


0818 


o6l I 


30 


1469 


0937 


0532 


40 


I50S 


I055 


^453 


50 


1543 


1174 


0374 


34 °° 


1587 


1293 


O294 


10 


1627 


1414 


02I3 


20 


1667 


1532 


OI35 


30 


1707 


1652 


0055 


40 


1746 


1769 


O.OOI9977 


50 


1735 


1889 


9896 


35 00 


1828 


2014 


9314 


10 


1868 


2134 


9734 


20 


1909 


2255 


9654 


30 


1949 


2376 


9573 


40 


2989 


2499 


9490 


50 


2029 


2619 


9410 


36 00 


2070 


2743 


9327 


10 


2III 


2865 


9246 


20 


2152 


2987 


9165 


30 


2192 


3HO 


9082 


40 


2233 


3234 


8999 


50 


2274 


3354 


8920 


37 °° 


2316 


3480 


8836 


10 


2358 


3602 


3756 


20 


2398 


3727 


8671 


30 


244O 


3851 


8589 


40 


2482 


3975 


8507 


50 


2523 


4098 


8425 


38 00 


2565 


4225 


3 340 


10 


2607 


4350 


8257 


20 


2648 


4475 


8i73 


30 


269O 


4599 


8091 


40 


2732 


4726 


8006 


50 


2775 


4846 


7929 


39 °° 


2815 


4977 


7S38 


10 


2857 


5102 


7755 


20 


2899 


5228 


7671 


30 


294I 


5355 


7536 


40 


2984 


5482 


7502 


50 


3025 


5608 


7417 



FORMULAE AND FACTORS. 



263 





v- a 


R _ *(i-, 2 ) 




Lat. 


(1 -*- 2 sin 2 L)\' 


(1 - e*- sin 2 Xj|' 


Log (i + <r 2 cos 2 Z). 




Log N. 


Log/?. 




40°oo' 


6.8053068 


6.8035734 


O.OOI7334 


IO 


3ITI 


5&59 


7252 


20 


3154 


59S7 


7167 


SO 


3195 


6lI5 


7080 


40 


3237 


6242 


6995 


50 


32S0 


6367 


6913 


41 OO 


3321 


6497 


6S24 


IO 


3365 


6625 


6740 


20 


3407 


6752 


6655 


30 


3450 


6SSO 


65/O 


40 


3592 


700S 


64S4 


50 


3535 


7I30 


6405 


42 OO 


3577 


7263 


6294 


IO 


3620 


7392 


6228 


20 


3663 


7519 


6144 


30 


3706 


7649 


6057 


40 


3749 


7777 


5972 


50 


3792 


79°5 


5SS7 


43 00 


3332 


8032 


5802 


10 


3S77 


8160 


5717 


20 


3919 


8288 


5631 


30 


3962 


8417 


5545 


40 


4004 


8549 


5455 


50 


4047 


8680 


5367 


44 00 


4090 


8803 


5287 


10 


4134 


8930 


5204 


20 


4177 


9059 


5116 


30 


4219 


9188 


5031 


40 


4262 


9317 


4945 


50 


4306 


9445 


4861 


45 00 


4347 


9575 


4772 


10 


439 1 


9704 


4687 


20 


4434 


9834 


4600 


30 


4477 


9961 


45i6 


40 


4519 


6 . 8040090 


4429 


50 


4563 


0218 


4345 


46 00 


4604 


0347 


4258 


TO 


4648 


0476 


4172 


20 


4690 


0605 


4085 


SO 


4734 


0734 


4000 


40 


4777 


0860 


3917 


50 


4S20 


0989 


3831 


47 00 


4861 


1118 


3744 


10 


4905 


1247 


3658 


20 


4948 


1376 


3572 


30 


499 1 


1504 


3487 


40 


5033 


1631 


3402 


50 


5076 


1759 


3317 



264 



GEODETIC OPERATIONS. 





(1 -*9sin a Z.)f 


«d - ,») 




Lat. 


(x - e^ sin 2 ZOf 


Lo & (1 + * 2 cos 2 Z). 


48°oo' 


LogiV. 


Log R. 




6.8055118 


6.8041887 


O.OOI3231 


10 


5l6o 


2016 


3144 


20 


5202 


2144 


3058 


30 


5244 


2272 


2972 


40 


5289 


2400 


2889 


50 


5333 


2528 


2805 


49 00 


5374 


2b 57 


2717 


10 


5417 


2784 


2633 


20 


5459 


2909 


2550 


30 


5501 


3037 


2464 


40 


5545 


3163 


2382 


50 


558V 


3293 


2294 


50 00 


5629 


3418 


2211 


10 


5672 


3544 


2128 


20 


5714 


3671 


2043 


30 


5756 


3798 


1958 


40 


5798 


3925 


1873 


50 


5841 


4048 


I79O , 



THE A, B, C, D, E GEODETIC FACTORS. 

From latitude 24 to 48 , inclusive. 

A= l 
B = 
C = 



Nave i"" 

1 
R arc \ ,r 

tan L 
2NR arcT 



I-*? 2 sin L cos L 
D = (1 — Vsin'Z)!' 

1 + 3 tan 8 Z 



Referred to Clarke's spheroid of 1866. 



FORMULA AND FACTORS. 



265 



Lat. 


Log A. 


Log£. 


LogC. 


Log I). 


Log E. 


24°oo' 


8. T 0948 34 


8. 5 i 19462 


I.05456 


2.2629 


5-8147 


OS 


818 


415 


625 


40 


59 


IO 


802 


368 


794 


52 


72 


15 


786 


320 


962 


64 


35 


20 


769 


271 


1. 06130 


75 


97 


25 


753 


223 


297 


86 


5.8210 


30 


738 


174 


464 


97 


23 


35 


720 


127 


631 


2.2708 


36 


40 


704 


078 


797 


19 


49 


45 


688 


028 


962 


30 


62 


50 


672 


8.5118979 


1. 07128 


40 


74 


55 


659 


Q30 


-93 


5i 


87 


25 00 


640 


882 


• 457 


62 


5.8300 


05 


623 


833 


621 


72 


13 


10 


607 


782 


735 


83 


26 


15 


59i 


733 


948 


93 


39 


20 


573 


684 


1.08111 


2 . 2804 


52 


25 


556 


634 


274 


15 


66 


30 


54i 


535 


435 


25 


79 


35 


524 


535 


597 


35 


92 


40 


508 


484 


759 


45 


5.84 5 


45 


491 


437 


920 


55 


18 


50 


473 


383 


1.09080 


65 


3i 


55 


456 


337 


241 


75 


45 


26 00 


440 


283 


400 


85 


58 


05 


423 


232 


560 


95 


7i 


10 


406 


181 


719 


2.2905 


85 


15 


388 


130 


878 


15 


93 


20 


372 


078 


1 . 10036 


24 


5-35I2 


25 


354 


027 


194 


34 


25 


30 


337 


8.5H7977 


352 


44 


39 


35 


320 


924 


509 


53 


52 


40 


303 


874 


666 


63 


66 


45 


287 


811 


854 


72 


79 


50 


270 


770 


979 


81 


93 


55 


25*2 


718 


I.HI35 


9i 


5 . 8606 


27 00 


235 


667 


290 


2 . 3000 


20 


05 


218 


616 


445 


09 


34 


10 


201 


564 


600 


18 


47 


15 


182 


5ii 


755 


27 


61 


20 


166 


458 


909 


36 


75 


25 


148 


405 


1. 12063 


45 


S 9 


30 


132 


353 


217 


54 


5.8702 


35 


113 


310 


37o 


63 


16 


40 


095 


248 


523 


72 


30 


45 


077 


195 


676 


Si 


44 


50 


059 


141 


828 


89 


58 


55 


041 


089 


980 


98 


69 



266 



GEODETIC OPERATIONS. 



Lat. 


Log A. 


Log B. 


LogC 


LogZ>. 


Log E. 


28°oo' 


8.5094025 


8. 5 1 17036 


I.13132 


2.3107 


5.8786 


05 


006 


8.5116983 


284 


15 


99 


10 


8.5093989 


930 


435 


24 


5-8813 


15 


970 


876 


586 


32 


27 


20 


952 


823 


737 


41 


4i 


25 


936 


768 


887 


49 


56 


30 


918 


715 


1. 14037 


57 


70 


35 


899 


661 


187 


65 


84 


40 


881 


608 


336 


74 


98 


45 


863 


552 


485 


82 


5.8912 


50 


845 


498 


634 


90 


26 


55 


827 


444 


783 


98 


40 


29 00 


808 


39° 


932 


2 . 3206 


55 


05 


790 


335 


1. 15080 


14 


69 


10 


772 


281 


227 


22 


83 


15 


753 


226 


375 


29 


98 


20 


735 


171 


522 


37 


5.9012 


25 


716 


116 


669 


45 


26 


30 


698 


061 


816 


53 


41 


35 


679 


007 


9 6 3 


60 


55 


40 


661 


8.5II5950 


1.16109 


63 


70 


45 


644 


896 


255 


75 


84 


50 


624 


841 


401 


83 


98 


55 


605 


7S7 


546 


90 


5.9H3 


30 00 


588 


728 


691 


98 


27 


05 


57o 


672 


835 


2.3305 


42 


10 


552 


616 


9S1 


12 


57 


15 


533 


561 


1.17126 


19 


7i 


20 


514 


505 


270 


27 


86 


25 


494 


449 


414 


34 


5.9201 


30 


476 


394 


558 


4i 


15 


35 


458 


337 


701 


48 


30 


40 


439 


280 


845 


55 


45 


45 


420 


225 


988 


62 


60 


50 


401 


168 


1.18131 


69 


74 


55 


376 


112 


274 


75 


89 


31 00 


361 


054 


416 


83 


5.9304 


05 


339 


8. 5 1 14998 


578 


89 


19 


10 


324 


942 


700 


96 


34 


15 


305 


884 


842 


2 . 3402 


49 


20 


286 


826 


984 


09 


64 


25 


267 


769 


1.19125 


16 


78 


30 


248 


712 


266 


22 


93 


35 


229 


655 


407 


29 


5 . 9408 


40 


211 


598 


548 


35 


23 


45 


192 


539 


688 


4i 


39 


50 


173 


4S3 


829 


48 


54 


55 


153 


424 


969 


54 


69 



FORMULAE AND FACTORS. 



267 



Lat. 


Log A. 


LogB. 


Log a 


LogZ>. 


LogE. 


32°oo' 


8.5093134 


8. 5 1 14367 


1. 20109 


2 . 3460 


5-9484 


05 


115 


309 


248 


66 


99 


10 


096 


251 


388 


73 


5-95 


4 


J 5 


077 


193 


527 


79 


29 


20 


057 


135 


666 


85 


44 


25 


037 


077 


805 


9i 


60 


30 


018 


020 


944 


97 


75 


35 


8.5092998 


8.5113963 


1. 21082 


2.3503 


90 


40 


979 


9°3 


221 


09 


5 . 9606 


45 


960 


844 


359 


14 


21 


50 


940 


786 


497 


20 


36 


55 


921 


727 


635 


26 


5i 


33 00 


901 


669 


772 


32 


67 


05 


881 


611 


910 


37 


82 


10 


862 


552 


1 . 22047 


43 


98 


15 


842 


492 


184 


48 


5-9713 


20 


823 


434 


321 


54 


29 


25 


803 


374 


453 


59 


44 


30 


783 


315 


594 


65 


60 


35 


764 


257 


730 


70 


75 


40 


744 


197 


867 


76 


9i 


45 


724 


137 


1 . 23003 


81 


5.9807 


50 


704 


078 


139 


86 


22 


55 


684 


018 


274 


91 


38 


34 00 


665 


8.5112959 


409 


97 


54 


05 


645 


898 


545 


2 . 3602 


69 


10 


625 


839 


680 


07 


85 


15 


605 


779 


815 


12 


5.9901 


20 


535 


720 


950 


17 


17 


25 


565 


660 


1.24085 


22 


32 


30 


545 


600 


220 


27 


48 


35 


525 


540 


353 


32 


64 


40 


505 


481 


489 


37 


80 


45 


485 


420 


623 


4i 


96 


50 


465 


363 


757 


46 


6.0012 


55 


445 


299 


891 


5i 


27 


35 00 


424 


238 


1.25023 


56 


44 


05 


404 


178 


157 


60 


60 


10 


383 ■ 


118 


290 


65 


76 


15 


363 


058 


424 


69 


92 


20 


344 


8.5111997 


557 


74 


6.0108 


25 


320 


936 


690 


78 


23 


30 


303 


875 


823 


83 


40 


35 


283 


814 


955 


87 


56 


40 


263 


753 


1.26088 


92 


72 


45 


243 


693 


220 


96 


88 


50 


223 


633 


353 


2.3700 


6 . 0204 


55 


203 


57i 


485 


04 


21 



268 



GEODETIC OPERATIONS. 



Lat. 


Log A. 


LogB. 


LogC. 


Log I). 


Log E. 


36 b oo' 


8.5092182 


8.5111509 


I. 2661 7 


2.3709 


6.0237 


05 


161 


448 


749 


13 


53 


10 


141 


387 


881 


17 


69 


15 


121 


326 


1. 27013 


21 


86 


20 


IOO 


265 


145 


25 


6 . 0302 


25 


080 


203 


276 


29 


18 


30 


060 


142 


407 


33 


35 


35 


039 


080 


539 


37 


51 


40 


Ol8 


Ol8 


670 


41 


67 


45 


8.509I998 


8.5IIO957 


801 


44 


84 


50 


978 


895 


931 


48 


6 . 0400 


55 


956 


834 


1.28062 


52 


17 


37 00 


930 


772 


193 


56 


33 


05 


915 


7IO 


323 


60 


50 


10 


894 


648 


454 


63 


66 


15 


874 


587 


584 


66 


83 


20 


854 


525 


714 


70 


6.0500 


25 


833 


462 


845 


74 


16 


30 


812 


401 


975 


77 


33 


35 


79I 


339 


1. 29104 


81 


50 


40 


771 


276 


234 


84 


66 


45 


750 


215 


364 


87 


83 


, 50 


729 


151 


494 


91 


6 . 0600 


55 


708 


090 


623 


94 


17 


38 00 


687 


027 


753 


97 


33 


05 


667 


8.5109964 


882 


2.3800 


50 


10 


646 


902 


1. 3001 1 


03 


67 


15 


625 


840 


140 


07 


84 


20 


604 


777 


269 


09 


6.0701 


25 


583 


715 


398 


13 


18 


30 


562 


652 


527 


16 


35 


35 


541 


59° 


656 


18 


5i 


40 


521 


526 


785 


22 


68 


45 


499 


463 


913 


24 


85 


50 


479 


401 


1. 3 1042 


27 


6.0802 


55 


458 


338 


170 


30 


19 


39 °° 


437 


275 


299 


33 


37 


05 


416 


212 


427 


35 


54 


10 


395 


150 


555 


38 


7i 


15 


374 


099 


683 


4i 


88 


20 


353 


023 


811 


43 


6 . 0905 


25 


332 


8.5108960 


939 


46 


22 


30 


3ii 


897 


1.32067 


48 


40 


35 


290 


843 


195 


51 


57 


40 


269 


770 


323 


53 


74 


45 


248 


707 


45o 


56 


91 


50 


227 


644 


578 


58 


6.1009 


55 


206 


58i 


706 


61 


26 



FORMULAE AND FACTORS. 



269 



Lat. 


Log A. 


LogB. 


Log C. 


LogZ>. 


Log,?. 


4o°oo' 


8.5091184 


8.5108518 


I.32833 


2.3863 


6.IO43 


05 


163 


455 


960 


65 


61 


10 


142 


393 


I.33088 


67 


78 


15 


125 


327 


215 


69 


96 


20 


099 


264 


342 


72 


6.II13 


25 


079 


201 


470 


74 


30 


30 


057 


137 


596 


76 


48 


35 


036 


073 


723 


78 


65 


40 


OI5 


010 


850 


80 


83 


45 


8 . 5090998 


8.5107946 


977 


82 


6.I20I 


50 


972 


883 


1. 34 104 


84 


18 


55 


952 


820 


231 


86 


36 


41 00 


930 


755 


358 


88 


54 


05 


909 


691 


485 


90 


71 


10 


888 


628 


611 


9i 


89 


15 


867 


574 


738 


93 


6.1307 


20 


845 


500 


864 


95 


24 


25 


824 


437 


991 


96 


42 


30 


803 


373 


I-3SII7 


98 


60 


35 


78i 


308 


244 


2.3900 


73 


40 


760 


244 


370 


01 


96 


45 


739 


181 


497 


03 


6.1413 


50 


7i8 


117 


623 


04 


3i 


55 


696 


o53 


749 


06 


49 


42 00 


675 


8. -106989 


874 


07 


67 


05 


653 


925 


1 . 36001 


08 


85 


10 


632 


861 


127 


10 


6.1503 


15 


610 


797 


253 


11 


21 


20 


590 


733 


379 


12 


39 


25 


568 


668 


5o5 


H 


57 


30 


547 


604 


631 


15 


75 


35 


524 


54i 


757 


16 


94 


40 


504 


476 


883 


17 


6.1612 


45 


483 


413 


i.370 9 


18 


30 


50 


460 


348 


135 


19 


48 


55 


439 


284 


261 


20 


66 


43 00 


419 


220 


386 


21 


85 


05 


396 


156 


512 


22 


6.1703 


10 


376 


092 


638 


23 


21 


15 


354 


028 


764 


24 


40 


20 


333 


8.5105963 


889 


25 


58 


25 


312 


899 


1. 38015 


25 


76 


30 


290 


835 


141 


26 


95 


35 


269 


771 


266 


27 


6.1813 


40 


247 


706 


392 


27 


32 


45 


226 


642 


5i8 


28 


50 


50 


204 


578 


643 


29 


69 


55 


183 


513 


769 


29 


87 



270 



GEODETIC OPERATIONS. 



Lat. 


Log A. 


-LogB. 


Log C, 


LogZ>. 


Log^:. 


44°oo' 


8.5090162 


8.5105449 


I.38894 


2 . 3930 


6.1906 


05 


140 


375 


I . 39020 


30 


24 


10 


Il8 


3ii 


145 


31 


43 


15 


097 


256 


271 


31 


62 


20 


076 


193 


396 


32 


80 


25 


054 


128 


522 


32 


99 


30 


033 


063 


647 


32 


6.2017 


35 


Oil 


8.5104999 


773 


32 


36 


40 


8 . 5089990 


935 


998 


33 


55 


45 


969 


870 


1.40024 


33 


74 


50 


947 


806 


149 


33 


93 


55 


925 


741 


275 


• 33 


6.2112 


45 00 


904 


677 


400 


33 


3i 


05 


883 


612 


526 


33 


50 


10 


861 


548 


651 


33 


69 


15 


840 


484 


777 


33 


88 


20 


818 


419 


902 


33 


6.2207 


25 


797 


356 


1. 41028 


33 


26 


30 


776 


291 


153 


33 


45 


35 


754 


226 


279 


33 


64 


40 


733 


162 


404 


33 


83 


45 


711 


098 


53o 


32 


6 . 2302 


50 


690 


034 


655 


32 


21 


55 


668 


8.5103969 


781 


32 


40 


46 00 


647 


905 


906 


31 


60 


05 


625 


841 


1.42032 


31 


79 


10 


604 


776 


157 


30 


98 


15 


583 


712 


283 


30 


6.2417 


20 


561 


648 


409 


29 


37 


25 


539 


584 


534 


29 


56 


30 


518 


518 


660 


28 


76 


35 


497 


457 


786 


28 


95 


40 


475 


392 


911 


27 


6.2514 


45 


454 


326 


1.43037 


26 


34 


50 


43i 


262 


163 


26 


53 


55 


410 


199 


289 


25 


73 


47 00 


390 


134 


414 


24 


93 


05 


368 


070 


539 


23 


6.2612 


10 


347 


005 


666 


22 


32 


15 


326 


8.5102941 


792 


21 


52 


20 


304 


876 


917 


21 


7i 


25 


283 


813 


1.44043 


20 


9i 


30 


261 


749 


169 


19 


6.2711 


35 


240 


685 


295 


17 


30 


40 


219 


621 


421 


16 


50 


45 


197 


557 


547 


15 


70 


50 


176 


493 


673 


14 


90 


55 


155 


428 


799 


13 


6.2810 



FORMULA*. AND FACTORS. 



271 



Lat. 


Log A . 


Log B. 


LogC. 


LogZ?. 


Log,E. 


48°oo' 


8.5089133 


8.5102364 


I.44926 


2.3912 


6.2830 


05 


112 


30I 


1.45052 


IO 


50 


10 


09I 


236 


178 


09 


70 


15 


070 


172 


304 


08 


90 


20 


O48 


IOS 


431 


06 


6.2910 


25 


O27 


045 


557 


OS 


30 


30 


OO5 


8.5101981 


683 


03 


50 


35 


8.5088984 


917 


809 


02 


70 


40 


963 


853 


937 


OO 


91 


45 


941 


789 


1 . 46063 


2.3899 


6.3011 


50 


920 


725 


189 


97 


31 


55 


899 


662 


316 


95 


51 


49 00 


878 


593 


442 


94 


72 



272 



GEODETIC OPERATIONS. 



FROM UNITED STATES COAST SURVEY REPORT. 



AUXILIARY TABLES FOR CONVERTING ARCS OF THE CLARKE ELLIPSOID INTO 

ARCS OF THE BESSEL ELLIPSOID. 

[All corrections are positive.] 





Corrections to dM. 




Arguments L' 


and dM. 




dM t 


0' 


5°' 


4°' 


3°' 


20' 


io' 


60" 


5°" 


40" 


30" 


20" 


10" 


5" 


Lat. 




























o 


n 


11 


11 


II 


// 


II 


11 


11 


11 


// 


11 


II 


II 


23 


481 


0.40T 


0.320 


O.24O 


0.160 


0.080 


0.008 


0.006 


0.005 


0.004 


0.003 


O.OOI 


0006 


24 


4S4 


•4°3 


.322 


.242 


.161 


.080 


.008 


.006 


.005 


.004 


• 003 


.001 


.0006 


25 


486 


.405 


.324 


•243 


.162 


.o8r 


.008 


.006 


.005 


.004 


.003 


.001 


.0006 


26 


489 


.407 


.326 


•245 


.163 


.081 


.008 


.006 


.005 


.004 


.003 


.001 


.0006 


27 


4Q1 


•409 


•327 


.246 


. 164 


.082 


.008 


.006 


.005 


.004 


.003 


.001 


.0006 


28 


494 


.411 


■3 2 9 


•247 


.165 


.082 


.008 


.007 


.005 


.004 


.003 


.001 


.0006 


29 


496 


•4i3 


•330 


.248 


.166 


.083 


.008 


.007 


.005 


.004 


.003 


.001 


.0006 


30 


407 


.416 


•332 


.250 


.167 


.083 


.008 


.007 


.005 


.004 


• 003 


.001 


.0006 


3 1 


502 


.418 


• 334 


.25I 


.168 


.084 


.008 


.007 


.006 


.004 


.003 


.001 


.0006 


32 


505 


.420 


•336 


•253 


.169 


.084 


.008 


.007 


.006 


.004 


.003 


.001 


.0006 


33 


507 


.422 


•338 


.254 


.169 


.085 


.008 


.007 


.006 


.004 


.003 


.001 


.0006 


34 


5io 


•425 


•340 


•255 


.170 


.085 


.008 


.007 


.006 


.004 


.003 


.001 


.0006 


35 


5'3 


•427 


•342 


.256 


.171 


.086 


.008 


.007 


.006 


.004 


.003 


.001 


.0006 


36 


516 


•43o 


•342 


.258 


.172 


.086 


.009 


.007 


.006 


.004 


.003 


.001 


.0006 


37 


518 


■432 


•345 


•259 


• i73 


.087 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


38 


521 


•434 


•347 


.26l 


.174 


.087 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


39 


524 


■43 6 


•349 


.262 


•i75 


.088 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


40 


527 


•439 


■35' 


.264 


.176 


.088 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


41 


530 


• 44 1 


•353 


.265 


.177 


.089 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


42 


533 


•444 


■355 


.267 


.178 


.089 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


43 


536 


.446 


•357 


.268 


.179 


.090 


.009 


.007 


.006 


.004 


.003 


.001 


.0007 


44 


539 


•449 


•359 


.27O 


.180 


.090 


.009 


.008 


.006 


.005 


.003 


.001 


.0007 


45 


542 


0.451 


0.361 


O.27I 


0.181 


o.ogr 


0.009 


0.008 


0.006 


0.005 


0.003 


001 


0007 






Co 


rrectio 


ns to a 


L. 






Ar 


jumen 


2 


V 

— an 


i dL. 




dL. 6 


0' 


50' 


40' 


3°' 


2o' 


io' 


60" 


50" 


40" 


30" 


20" 


10" 


5" 


Lat. 































11 


// 


// 


a 


// 


// 


11 


// 


// 


11 


11 


11 


II 


23 


193 


0.160 


0.129 


0.096 


O 064 


O.O32 


0.003 


0.003 


0.002 


0.002 


O.OOI 


O.OOI 


. 0003 


24 


20c 


.165 


• 133 


.099 


.066 


•°33 


.003 


.003 


.002 


.002 


.001 


.001 


.0003 


25 


206 


.171 


.138 


.103 


.068 


•034 


.003 


.003 


.002 


.002 


.001 


.001 


.0003 


26 


213 


•177 


.142 


.106 


.070 


•o35 


.003 


.003 


.002 


.002 


.001 


.001 


.0003 


27 


220 


.183 


.147 


.110 


•073 


•037 


.004 


.003 


.002 


.002 


.001 


.oot 


.0003 


28 


227 


.189 


•I5 1 


.113 


.075 


.038 


.004 


.003 


.002 


.002 


.001 


.001 


. 0003 


29 


234 


. 196 


.156 


.117 


.078 


•o39 


.004 


.003 


.002 


.002 


.001 


.001 


.0003 


30 


242 


.202 


.161 


. J2I 


.080 


.040 


.004 


.003 


.002 


.002 


.001 


.001 


.0003 


3 1 


2 so 


.2oq 


.167 


.125 


.083 


.042 


.004 


.003 


.003 


.002 


.001 


.001 


.0004 


32 


258 


.216 


.172 


.129 


.086 


•043 


.004 


.003 


.003 


.002 


.001 


.001 


.0004 


33 


207 


.22? 


.178 


•133 


.089 


•045 


.005 


.003 


.003 


.002 


.002 


.001 


.0004 


34 


275 


.230 


.184 


•137 


.091 


.046 


.005 


.003 


.003 


.002 


.002 


.001 


.0004 


35 


283 


•237 


.190 


.141 


.094 


.047 


.005 


.004 


.003 


.002 


.002 


.001 


.0004 


36 


291 


.243 


•195 


.145 


•097 


.048 


.005 


.004 


.003 


.002 


.002 


.001 


.0004 


37 


300 


.250 


.201 


.150 


.IOO 


.050 


.005 


.004 


.003 


.002 


.002 


.oot 


.0004 


38 


30.3 


•257 


.206 


•154 


.IO3 


.051 


.005 


.004 


.003 


.002 


.002 


.001 


.0004 


39 


3 r 7 


.264 


.212 


.158 


.106 


.053 


.005 


.004 


.004 


.003 


.002 


.001 


.0004 


40 


3^5 


271 


.217 


.162 


.108 


.054 


.005 


.004 


.004 


.003 


.002 


.001 


.0005 


41 


334 


.278 


.223 


.167 


.III 


.056 


.006 


.004 


.004 


.003 


.002 


.001 


.0005 


42 


343 


.236 


.229 


.171 


.114 


•057 


.006 


.004 


.004 


.003 


.002 


.001 


.0005 


43 


352 


• 294 


.236 


.176 


.117 


.059 


.006 


.005 


.004 


.003 


.002 


.001 


.0005 


44 


302 


.302 


.242 


.181 


.120 


.060 


.006 


.005 


.004 


.003 


.002 


.001 


.0005 


\ 45 


372 


0.310 


O.249 


0.186 


O.I24 


0.062 


0.006 


0.005 


0.004 


0.003 


0.002 


O.OOI 


0.0005 



FORMULAE AND FACTORS. 



2/3 



TAKEN FROM U. S. COAST AND GEODETIC SURVEY REPORT. 

SUBSIDIARY TABLE FOR REFERRING VALUES OF COEFFICIENTS A, B, C, D, E, 
FROM CLARKE'S TO BESSEL's ELLIPSOID. 



Lat. 


To log A add. 


To log B add. 


To log Cadd. 


From log D 
subtract. 


To log E add. 


23° 


O. OOOO5 S 2 


O.OOOO233 


O . OOOO8 


. 00G1 


O.OOOI 


24 


584 


241 


08 


61 




25 


537 


249 


08 


6l 




26 


590 


258 


08 


6l 




27 


593 


266 


09 


6l 




2S 


596 


274 


09 


6l 




29 


599 


283 


09 


6l 




30 


602 


293 


09 


6l 




31 


605 


302 


09 


6l 




32 


609 


312 


09 


6l 




33 


612 


321 


09 


6l 




34 


615 


331 


09 


6l 




35 


619 


342 


10 


6l 




36 


622 


352 


IO 


6l 




37 


625 


362 


IO 


6l 




33 


629 


372 


IO 


6l 




39 


632 


383 


IO 


6l 




40 


636 


393 


IO 


6l 




4i 


639 


404 


IO 


6l 




42 


643 


415 


II 


6l 




43 


647 


425 


II 


6l 




44 


650 


436 


II 


6l 




45 


654 


447 


II 


6l 




46 


657 


453 


II 


6l 




47 


661 


468 


II 


6l 




4 s 


664 


479 


II 


6l 




49 


668 


490 


12 


6l 




50 


672 


501 


12 


6l 





TABLE OF LOG F. 



Lat. 



23 
24 
25 
26 

27 
28 
29 



LogF. 

7.812 
23 
32 
41 
49 
- 55 
61 



Lat. 



SO' 

31 

32 

33 

34 

35 

36 



Log F. 



7.866 
70 

73 
75 
77 

77 
77 



Lat. 



37 J 

33 

39 

40 

41 

42 

43 



Log^. 



,876 
74 
72 
69 

64 
60 

54 



Lat. 



LogF. 



44° 


7.848 


45 


40 


46 


32 


47 


24 


48 


14 


49 


04 


50 


7.792 



274 



GEODETIC OPERATIONS. 



TABLE OF CORRECTIONS TO LONGITUDE FOR DIFFERENCE 
IN ARC AND SINE. 



Log K{-). 


Log difference. 


Log dM(+). 


Log K(-). 


Log difference. 


Log dM{-±). 


3.871 


O. OOOOOO I 


2.380 


4.9I3 


O.OOOOII9 


3-422 


3 


970 


002 


2.479 


4.922 


124 


3 


•431 


4 


•115 


003 


2.624 


4.932 


130 


3 


.441 


4 


.171 


OO4 


2.680 


4.941 


136 


3 


.450 


4 


.221 


005 


2.730 


4-950 


142 


3 


459 


4 


268 


006 


2.777 


4-959 


147 


3 


468 


4 


292 


OO7 


2.8oi 


4.968 


153 


3 


477 


4 


309 


008 


2.818 


4.976 


160 


3 


485 


4 


320 


009 


2.839 


4-985 


166 


3 


•494 


4 


361 


OIO 


2.870 


4-993 


172 


3 


502 


4 


383 


on 


2.892 


5.002 


179 


3 


5ii 


4 


415 


012 


2.924 


5.010 


186 


3 


519 


4 


430 


013 


2.939 


5-017 


192 


3 


526 


4 


445 


OT4 


2-954 


5-025 


199 


3 


534 


4 


459 


015 


2.968 


5-Q33 


206 


3 


542 


4 


473 


016 


2.982 


5-040 


213 


3 


549 


4 


487 


017 


2.996 


5-047 


221 


3 


556 


4 


500 


018 


3-OOg 


5-o54 


228 


3 


563 


4 


524 


020 


3-033 


5.062 


236 


3 


57i 


4 


548 


023 


3-057 


5.068 


243 


3 


577 


4 


57o 


025 


3-079 


5 -075 


251 


3 


584 


4 


59i 


027 


3.100 


5.082 


259 


3 


59i 


4 


612 


030 


3. 121 


5.088 


267 


3 


597 


- 4 


631 


033 


3.I4O 


5-095 


275 


3 


604 


4 


649 


036 


3-158 


5.102 


284 


3 


611 


4 


667 


039 


3.I76 


5.108 


292 


3 


617 


4 


684 


042 


3-^93 


5- 114 


300 


3 


623 


4 


701 


045 


3.2IO 


5.120 


309 


3 


629 


4 


716 


048 


3-225 


5.126 


318 


3 


635 


4 


732 


052 


3.241 


5-132 


327 


3 


641 


4 


746 


056 


3-255 


5.138 


336 


3 


647 


4 


761 


059 


3.270 


5-144 


345 


3 


653 


4 


774 


063 


3.2S3 


5.150 


354 


3 


659 


4 


788 


067 


3 297 


5.156 


364 


3 


665 


4 


8or 


071 


3-3io 


5- 161 


373 


3 


670 


4 


813 


075 


3-322 


5-I67 


383 


3 


676 


4 


825 


080 


3-334 


5-172 


39 2 


3 


681 


4 


834 


0S4 


3-343 


5.178 


402 


3 


687 


4 


849 


089 


3-358 


5-183 


412 


3 


692 


4 


860 


094 


3.369 


5.188 


422 


3 


697 


4 


871 


098 


3-3So 


5-193 


433 


3 


702 


4 


882 


103 


3-39 1 


5-199 


443 


3 


708 


4 


892 


108 


3.401 


5.204 


453 


3- 


713 


4-903 


114 


3.412 









NDEX 



PAGE 

Abulfeda, description of Arabian arc-measurement, 3 

Adjusting the azimuth 200 

Adjustment, figure 168 

station 146 

when directions have been observed 180 

Airy 248 

Alexandria 2 

Anaximander , I 

Angles, method of measuring 97 

Arabian arc-determination 2 

unit of measure 28 

Arago 15 

Argelander 18 

Auzout 29 

Axes of the earth, Clarke's and Bessel's values of 259 

Azimuth, affected by adjustment 200 

formula for computing 218 

Baeyer 20 

Barrow, Indian arc-measurement 13 

Base apparatus, first form of 50 

requisites of 50 

Bache-Wurdeman , 52 

Baumann. ... 60 

Bessel 58 

Borda 51 

Colby 59, 64 

Ibafiez 60 

Lapland 50 

Peru 50 

Porro 59 

Repsold 60 

Struve 59 



2/6 INDEX. 

PAGE 

Base-line, probable error of 74 

reduction to sea-level 78 

Base-measurements 49 

aligning 64 

comparison of results 61 

computation of results 70 

erection of terminal marks 66 

general precautions 69 

inclination 68 

instructions 64 

sector error 68 

selecting site 64 

record, form of 66 

references 79 

transferrence of end to the ground 67 

Beccaria, arc measurement 10 

Bessel 247 

base apparatus 20 

review of the French arc 15 

Biot 15 

Bonne 19 

Borda, metallic thermometer 14 

Borden, survey of Massachusetts 23 

Boscovich, arc-measurement 10 

Bouguer 7 

Boutelle 81,86 

Brahe, Tycho 29 

Briggs 29 

Caldewood, glass base apparatus 11 

Camus 8 

Cape of Good Hope arc 10 

Cassini 4, II 

revision of the French arc 9 

Celsius 8 

Centre, reduction to 196 

Chaldean unit of measure 28 

Chauvenet , . . . , 160 

Circle, entire, first used 29 

Clairault 8 

theory of the figure of the earth 9 



INDEX. 277 

PAGE 

Clarke, solution for the figure of the earth 248 

reference to the great English theodolite 12 

Coast Survey, U. S. , organized 23 

form of base apparatus 52, 61 

heliotrope 46 

signals 84 

theodolite ... 32 

Colonna go 

Commission for European degree-measurements 26 

Comparison of base-bars with a standard 71 

Condamine, De la , 7 

Connection of France and England by triangulation 11 

Constants and their logarithms, table of 260 

Correction for inclination 74 

Correlatives, equations of 159 

Cutts 86 

Davidson 47, 102 

Delambre 13 

revision of the Peruvian arc 8 

Des Hays, pendulum-investigations 6 

Directions, adjustment of . 180 

horizontal, copy of record 101 

Dividing-engine first used 3° 

Dixon. See Mason. 

Doolittle 195 

Eccentric signal 146 

instrument 196 

Ellipticity of the earth, table of 249 

Errors, mean of 146 

probable. See Probable Errors. 

Equations, conditional, number of , 179 

side 171 

solution of, by logarithms 194 

Eratosthenes 1 

Everest, Indian arc 14 

Expansion coefficient, determination of 72 

Fernel 3 

Figure-adjustment 168 

Figure of the earth 234 

literature of 249 



278 INDEX. 

PAGE 

French Academy arc-measurements in Lapland and Peru 6 

Froriep 1 

Gascoigne, first to use spider-lines 4 

Gauss 19, 159 

Geodesie, Ecole speciale de 17 

Geodetic factors, tables of 264 

God in 7 

Greek unit of measure 2S 

H ANSTEEN l8 

Heights determined by barometer 94 

triangulation ... 101 

Heliotrope, description of 45 

first used 45 

illustration 46 

use and adjustments 46 

Hilgard 41 

Hipparchus 29 

Hounslovv Heath base . 12 

Humboldt 29 

Huygens's theory of centrifugal motion 6 

Ingenieur Corps, organization of 16 

Instruments 28 

Invention of the vernier 29 

Isle, De 1' 17 

Italian commission organized ; 25 

Italy, co operation with Switzerland 17 

James, Sir Henry, inference to the great English theodolite 11 

Lacaille, revision of Picard's arc 4 

arc-measurement at the Cape of Good Hope 10 

Lahire 4 

Lambton, Indian arc 1 1 

Lapiace 13, 247 

Lapland arc-measurement 8 

Latitude, formula for computing 203 

illustration 220, 222 

Least squares, theory of 104 



INDEX. 279 

PAGE 

Legendre 11,13 

Letronne, review of Posidonius's arc 2 

Level, table giving difference between the true and apparent 92 

Liesganig, arc-measurement 10 

Litirow 31 

Longitude determined by powder-flashes 19 

Longitude, formula for computing 217 

illustrations 220, 223 

Maclear, continuance of the Cape of Good Hope arc 23 

Maraldi 4 

Mason, Maryland-Pennsylvania boundary-line 10 

Maupertuis, Lapland arc 8 

Mayer, repeating-theodolites 12 

Mechain 11,15 

Metre, determination of its length 13 

legal and recent values of 249 

Metres to feet, table for converting 257 

miles 25S 

yards 257 

Micrometer first used 29 

determination of run 35 

Miles to metres, table for converting 259 

Monnier 8 

Mosman 97 

Mudge 15 

Muffling, Von : 19 

Musschenbroeck 19 

Napier 29 

Newton, theory of universal gravitation demonstrated by Picard's arc 4 

Norrr>al equations 155 

Normals, table of 261 

Norwood, measurement of the distance from London to York 4 

Nunez 29 



OUTHIER 



Palanber 14 

Pamlico-Chesapeake arc 24 

Phase, correction for « 144 



280 INDEX. 

PAGE 

Picard's triangulation 4 

Pcle, selection of, in figure-adjustment 179 

Posch, view of Ptolemy's length of a degree 2 

Posidonius, arc-measurement 2 

Probable error of the arithmetical mean 124 

of a single determination 120 

illustration 122 

of a base line „ 125 

direction 125 

in the computation of unknown quantities in triangles 127, 133, 

136, 137, 133 

Prussia, first geodetic work 16 

Prussian.-Russian connection 25 

Ptolemy, value of earth's circumference 2 

Puissant, review of the French arc 15 

Pythagoras I 

Quadrant of the earth, length of 249 

Radius of curvature, table of 261 

Reduction to centre 196 

Reichenbach 26, 31 

Repetition of angles, principle first announced 12 

abandoned 31 

Repsold 26, 31 

Riccioli 3 

Richer, expedition to Cayenne 6 

Roemer 29 

Roy 11, 15 

Russian arc, accuracy of 18 

Russia, first geodetic work 17 

Saegmuller, principal of bisection 31 

Schott 24, 179, 248 

Schmidt 247 

Schumacher 19, 31 

Schwerd 19 

Series, binomial 256 

exponential 256 

logarithmic 257 



INDEX. . 28l 

PAGE 

Signals, form used on coast-survey 84 

night cost of 82 

method of erection 87 

size and lengths of timbers 88 

Snellius' triangulation 3 

Spain, first geodetic work 25 

Spherical excess, computation of 168 

Speyer base 19 

Spider-lines in telescope, first use of 4 

Stations, description of 95 

permanent markings 96 

intervisibility of 90 

Station adjustment 146 

Struve 17 

Svanberg ; . 14 

Sweden, coast triangulation 26 

Swedish arc-measurement in Lapland 14 

Switzerland, first geodetic work 17 

Syene 1 

Tenner 17 

Thales I 

Theodolites, adjustment of 33 

construction of 32 

errors of eccentricity 36 

graduation 39 

illustration 34 

size of 32 

Toise of Peru 7 

Transferring underground mark to the top of a signal 96 

Triangles, best composition of 86 

Triangulation, calculation of 143 

conditions to be fulfilled 143 

Trigonometric expressions 253 

series 256 



Ulloa. 



7 



Varin, pendulum-investigations 6 

Vernerius 29 



282 INDEX. 

PAGE 

Vernier, invention of 29 

distance apart 37 

Walbeck 247 

Walker 22 

Waugh 22 

Weights, application of, in adjustments 165 

Yards to metres, table for converting 258 

Yollond : 184 

Zach, Von, revision of Beccaria's arc 10 

of the Peruvian arc 8 



9151 



i i 









■•••% y.tfKfcV AisakX y.-- % 



' °o 




*V V^\** v"V v^v 










0* .•!•.. *o 






/ ** % %'-, 



V •«.-.. 












.0* 









*• ^ v 











o° # <^* 



<v ^ 
















* -. . . 



♦-o* : 




.0«^ 



AZ 



• I *» * A U 3> # • N ' . $* 

01 • lils > v 

























»°^ 'life' ^ 

/ y\ °^w. : /\ 









* -v 






V 















•♦^V V-^V v^V <v^ f V 





£ -<>••- "^ 



<^ ♦7?rr^ a 



** v \ 








WERT 
BOOKBINDING 

MIOOLETOWN PA 

DEC 83 












v 
^ 



